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Properties

Label 8-294e4-1.1-c7e4-0-1
Degree 88
Conductor 74711820967471182096
Sign 11
Analytic cond. 7.11459×1077.11459\times 10^{7}
Root an. cond. 9.583389.58338
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s − 54·3-s + 64·4-s − 540·5-s − 864·6-s − 1.02e3·8-s + 729·9-s − 8.64e3·10-s − 1.19e3·11-s − 3.45e3·12-s − 2.46e4·13-s + 2.91e4·15-s − 1.63e4·16-s − 2.94e4·17-s + 1.16e4·18-s + 4.70e4·19-s − 3.45e4·20-s − 1.91e4·22-s + 9.40e4·23-s + 5.52e4·24-s + 2.26e5·25-s − 3.93e5·26-s + 3.93e4·27-s − 3.41e5·29-s + 4.66e5·30-s + 2.58e5·31-s − 6.55e4·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.93·5-s − 1.63·6-s − 0.707·8-s + 1/3·9-s − 2.73·10-s − 0.270·11-s − 0.577·12-s − 3.10·13-s + 2.23·15-s − 16-s − 1.45·17-s + 0.471·18-s + 1.57·19-s − 0.965·20-s − 0.383·22-s + 1.61·23-s + 0.816·24-s + 2.90·25-s − 4.39·26-s + 0.384·27-s − 2.59·29-s + 3.15·30-s + 1.56·31-s − 0.353·32-s + ⋯

Functional equation

Λ(s)=((24⋅34⋅78)s/2 ΓC(s)4 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
Λ(s)=((24⋅34⋅78)s/2 ΓC(s+7/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅34⋅782^{4} \cdot 3^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 7.11459×1077.11459\times 10^{7}
Root analytic conductor: 9.583389.58338
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅34⋅78, ( :7/2,7/2,7/2,7/2), 1)(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )

Particular Values

L(4)L(4) ≈\approx 0.017213020530.01721302053
L(12)L(\frac12) ≈\approx 0.017213020530.01721302053
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 (1−p3T+p6T2)2 ( 1 - p^{3} T + p^{6} T^{2} )^{2}
3C2C_2 (1+p3T+p6T2)2 ( 1 + p^{3} T + p^{6} T^{2} )^{2}
7 1 1
good5D4×C2D_4\times C_2 1+108pT+2596p2T2+304344p3T3+36012999p4T4+304344p10T5+2596p16T6+108p22T7+p28T8 1 + 108 p T + 2596 p^{2} T^{2} + 304344 p^{3} T^{3} + 36012999 p^{4} T^{4} + 304344 p^{10} T^{5} + 2596 p^{16} T^{6} + 108 p^{22} T^{7} + p^{28} T^{8}
11D4×C2D_4\times C_2 1+1196T−27613882T2−11876332624T3+435946651536427T4−11876332624p7T5−27613882p14T6+1196p21T7+p28T8 1 + 1196 T - 27613882 T^{2} - 11876332624 T^{3} + 435946651536427 T^{4} - 11876332624 p^{7} T^{5} - 27613882 p^{14} T^{6} + 1196 p^{21} T^{7} + p^{28} T^{8}
13D4D_{4} (1+12312T+99570968T2+12312p7T3+p14T4)2 ( 1 + 12312 T + 99570968 T^{2} + 12312 p^{7} T^{3} + p^{14} T^{4} )^{2}
17D4×C2D_4\times C_2 1+29484T+322588204T2−8077415824296T3−226187173694364513T4−8077415824296p7T5+322588204p14T6+29484p21T7+p28T8 1 + 29484 T + 322588204 T^{2} - 8077415824296 T^{3} - 226187173694364513 T^{4} - 8077415824296 p^{7} T^{5} + 322588204 p^{14} T^{6} + 29484 p^{21} T^{7} + p^{28} T^{8}
19D4×C2D_4\times C_2 1−47088T+902711442T2+22280872687488T3−1052880565778475733T4+22280872687488p7T5+902711442p14T6−47088p21T7+p28T8 1 - 47088 T + 902711442 T^{2} + 22280872687488 T^{3} - 1052880565778475733 T^{4} + 22280872687488 p^{7} T^{5} + 902711442 p^{14} T^{6} - 47088 p^{21} T^{7} + p^{28} T^{8}
23D4×C2D_4\times C_2 1−94052T+1475279102T2−52748942684816T3+12613095628633527907T4−52748942684816p7T5+1475279102p14T6−94052p21T7+p28T8 1 - 94052 T + 1475279102 T^{2} - 52748942684816 T^{3} + 12613095628633527907 T^{4} - 52748942684816 p^{7} T^{5} + 1475279102 p^{14} T^{6} - 94052 p^{21} T^{7} + p^{28} T^{8}
29D4D_{4} (1+170600T+41693255666T2+170600p7T3+p14T4)2 ( 1 + 170600 T + 41693255666 T^{2} + 170600 p^{7} T^{3} + p^{14} T^{4} )^{2}
31D4×C2D_4\times C_2 1−258984T−4480363142T2−4280447973029184T3+ 1 - 258984 T - 4480363142 T^{2} - 4280447973029184 T^{3} + 25 ⁣⋯ ⁣6725\!\cdots\!67T4−4280447973029184p7T5−4480363142p14T6−258984p21T7+p28T8 T^{4} - 4280447973029184 p^{7} T^{5} - 4480363142 p^{14} T^{6} - 258984 p^{21} T^{7} + p^{28} T^{8}
37D4×C2D_4\times C_2 1−648856T+135997011478T2−61740817520417152T3+ 1 - 648856 T + 135997011478 T^{2} - 61740817520417152 T^{3} + 32 ⁣⋯ ⁣8732\!\cdots\!87T4−61740817520417152p7T5+135997011478p14T6−648856p21T7+p28T8 T^{4} - 61740817520417152 p^{7} T^{5} + 135997011478 p^{14} T^{6} - 648856 p^{21} T^{7} + p^{28} T^{8}
41D4D_{4} (1+605556T+416076703796T2+605556p7T3+p14T4)2 ( 1 + 605556 T + 416076703796 T^{2} + 605556 p^{7} T^{3} + p^{14} T^{4} )^{2}
43D4D_{4} (1+698480T+535788255846T2+698480p7T3+p14T4)2 ( 1 + 698480 T + 535788255846 T^{2} + 698480 p^{7} T^{3} + p^{14} T^{4} )^{2}
47D4×C2D_4\times C_2 1+816912T−486667751318T2+114994027781212032T3+ 1 + 816912 T - 486667751318 T^{2} + 114994027781212032 T^{3} + 73 ⁣⋯ ⁣0373\!\cdots\!03T4+114994027781212032p7T5−486667751318p14T6+816912p21T7+p28T8 T^{4} + 114994027781212032 p^{7} T^{5} - 486667751318 p^{14} T^{6} + 816912 p^{21} T^{7} + p^{28} T^{8}
53D4×C2D_4\times C_2 1−1923508T+481825150322T2−1670827518121566544T3+ 1 - 1923508 T + 481825150322 T^{2} - 1670827518121566544 T^{3} + 46 ⁣⋯ ⁣2746\!\cdots\!27T4−1670827518121566544p7T5+481825150322p14T6−1923508p21T7+p28T8 T^{4} - 1670827518121566544 p^{7} T^{5} + 481825150322 p^{14} T^{6} - 1923508 p^{21} T^{7} + p^{28} T^{8}
59D4×C2D_4\times C_2 1−81864T−2292506761454T2+219239527631274432T3− 1 - 81864 T - 2292506761454 T^{2} + 219239527631274432 T^{3} - 92 ⁣⋯ ⁣8592\!\cdots\!85T4+219239527631274432p7T5−2292506761454p14T6−81864p21T7+p28T8 T^{4} + 219239527631274432 p^{7} T^{5} - 2292506761454 p^{14} T^{6} - 81864 p^{21} T^{7} + p^{28} T^{8}
61D4×C2D_4\times C_2 1+1616544T−1741805663448T2−3120682428761861952T3+ 1 + 1616544 T - 1741805663448 T^{2} - 3120682428761861952 T^{3} + 87 ⁣⋯ ⁣5987\!\cdots\!59T4−3120682428761861952p7T5−1741805663448p14T6+1616544p21T7+p28T8 T^{4} - 3120682428761861952 p^{7} T^{5} - 1741805663448 p^{14} T^{6} + 1616544 p^{21} T^{7} + p^{28} T^{8}
67D4×C2D_4\times C_2 1−5424T−2686337401526T2+51175745855801856T3− 1 - 5424 T - 2686337401526 T^{2} + 51175745855801856 T^{3} - 29 ⁣⋯ ⁣7329\!\cdots\!73T4+51175745855801856p7T5−2686337401526p14T6−5424p21T7+p28T8 T^{4} + 51175745855801856 p^{7} T^{5} - 2686337401526 p^{14} T^{6} - 5424 p^{21} T^{7} + p^{28} T^{8}
71D4D_{4} (1+2492964T+2402383804306T2+2492964p7T3+p14T4)2 ( 1 + 2492964 T + 2402383804306 T^{2} + 2492964 p^{7} T^{3} + p^{14} T^{4} )^{2}
73D4×C2D_4\times C_2 1+4710744T−4712548213920T2+22653309377619710928T3+ 1 + 4710744 T - 4712548213920 T^{2} + 22653309377619710928 T^{3} + 35 ⁣⋯ ⁣3535\!\cdots\!35T4+22653309377619710928p7T5−4712548213920p14T6+4710744p21T7+p28T8 T^{4} + 22653309377619710928 p^{7} T^{5} - 4712548213920 p^{14} T^{6} + 4710744 p^{21} T^{7} + p^{28} T^{8}
79D4×C2D_4\times C_2 1+3416656T−16415423029118T2−35255983728480310784T3+ 1 + 3416656 T - 16415423029118 T^{2} - 35255983728480310784 T^{3} + 19 ⁣⋯ ⁣1119\!\cdots\!11T4−35255983728480310784p7T5−16415423029118p14T6+3416656p21T7+p28T8 T^{4} - 35255983728480310784 p^{7} T^{5} - 16415423029118 p^{14} T^{6} + 3416656 p^{21} T^{7} + p^{28} T^{8}
83D4D_{4} (1−8921448T+74139859271942T2−8921448p7T3+p14T4)2 ( 1 - 8921448 T + 74139859271942 T^{2} - 8921448 p^{7} T^{3} + p^{14} T^{4} )^{2}
89D4×C2D_4\times C_2 1+12374532T+36847841483068T2+ 1 + 12374532 T + 36847841483068 T^{2} + 34 ⁣⋯ ⁣3634\!\cdots\!36T3+ T^{3} + 47 ⁣⋯ ⁣9947\!\cdots\!99T4+ T^{4} + 34 ⁣⋯ ⁣3634\!\cdots\!36p7T5+36847841483068p14T6+12374532p21T7+p28T8 p^{7} T^{5} + 36847841483068 p^{14} T^{6} + 12374532 p^{21} T^{7} + p^{28} T^{8}
97D4D_{4} (1+11296584T+159222144655440T2+11296584p7T3+p14T4)2 ( 1 + 11296584 T + 159222144655440 T^{2} + 11296584 p^{7} T^{3} + p^{14} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.19372717166763524443282543548, −7.00354555392393071973063247600, −6.96071209148334155809625143481, −6.48274049469276865503516575472, −6.19387862859329543480327377574, −5.96555986252891626465538329424, −5.41341351146950740890345396633, −5.27800530453057311643096538219, −5.07105035655427737807322838605, −4.83396623303813100283414273479, −4.67450372833506227838485553143, −4.63088673542699286657070684565, −4.22361973486608987298016253571, −3.67785906461735388962655685822, −3.64922001297478728360651975081, −3.20604284257515557751206280502, −2.96683272339681349683729663882, −2.65826338405347393626656597972, −2.50631735738321991031330012456, −1.99456571732652508579446822834, −1.44784042333171460711734877705, −1.12791331367024790688645648651, −0.71555204598066805016290251408, −0.13933595057218425724855472398, −0.06201554794410873072722200407, 0.06201554794410873072722200407, 0.13933595057218425724855472398, 0.71555204598066805016290251408, 1.12791331367024790688645648651, 1.44784042333171460711734877705, 1.99456571732652508579446822834, 2.50631735738321991031330012456, 2.65826338405347393626656597972, 2.96683272339681349683729663882, 3.20604284257515557751206280502, 3.64922001297478728360651975081, 3.67785906461735388962655685822, 4.22361973486608987298016253571, 4.63088673542699286657070684565, 4.67450372833506227838485553143, 4.83396623303813100283414273479, 5.07105035655427737807322838605, 5.27800530453057311643096538219, 5.41341351146950740890345396633, 5.96555986252891626465538329424, 6.19387862859329543480327377574, 6.48274049469276865503516575472, 6.96071209148334155809625143481, 7.00354555392393071973063247600, 7.19372717166763524443282543548

Graph of the ZZ-function along the critical line