Properties

Label 8-14e8-1.1-c7e4-0-2
Degree 88
Conductor 14757890561475789056
Sign 11
Analytic cond. 1.40535×1071.40535\times 10^{7}
Root an. cond. 7.824797.82479
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·3-s + 42·5-s + 894·9-s − 7.42e3·11-s − 2.36e4·13-s − 588·15-s + 1.57e4·17-s + 2.66e4·19-s − 3.26e4·23-s + 1.24e5·25-s + 2.64e4·27-s − 3.16e5·29-s − 1.80e5·31-s + 1.03e5·33-s + 4.58e4·37-s + 3.31e5·39-s + 6.43e5·41-s + 2.04e6·43-s + 3.75e4·45-s + 1.66e6·47-s − 2.21e5·51-s + 4.10e5·53-s − 3.11e5·55-s − 3.72e5·57-s + 1.70e6·59-s − 5.47e5·61-s − 9.93e5·65-s + ⋯
L(s)  = 1  − 0.299·3-s + 0.150·5-s + 0.408·9-s − 1.68·11-s − 2.98·13-s − 0.0449·15-s + 0.779·17-s + 0.890·19-s − 0.559·23-s + 1.59·25-s + 0.258·27-s − 2.40·29-s − 1.08·31-s + 0.503·33-s + 0.148·37-s + 0.894·39-s + 1.45·41-s + 3.92·43-s + 0.0614·45-s + 2.34·47-s − 0.233·51-s + 0.378·53-s − 0.252·55-s − 0.266·57-s + 1.07·59-s − 0.308·61-s − 0.448·65-s + ⋯

Functional equation

Λ(s)=((2878)s/2ΓC(s)4L(s)=(Λ(8s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
Λ(s)=((2878)s/2ΓC(s+7/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \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: 28782^{8} \cdot 7^{8}
Sign: 11
Analytic conductor: 1.40535×1071.40535\times 10^{7}
Root analytic conductor: 7.824797.82479
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 2878, ( :7/2,7/2,7/2,7/2), 1)(8,\ 2^{8} \cdot 7^{8} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )

Particular Values

L(4)L(4) \approx 2.9035526492.903552649
L(12)L(\frac12) \approx 2.9035526492.903552649
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
7 1 1
good3D4×C2D_4\times C_2 1+14T698T216240pT3490265p2T416240p8T5698p14T6+14p21T7+p28T8 1 + 14 T - 698 T^{2} - 16240 p T^{3} - 490265 p^{2} T^{4} - 16240 p^{8} T^{5} - 698 p^{14} T^{6} + 14 p^{21} T^{7} + p^{28} T^{8}
5D4×C2D_4\times C_2 142T123166T2+263088pT3+374647071p2T4+263088p8T5123166p14T642p21T7+p28T8 1 - 42 T - 123166 T^{2} + 263088 p T^{3} + 374647071 p^{2} T^{4} + 263088 p^{8} T^{5} - 123166 p^{14} T^{6} - 42 p^{21} T^{7} + p^{28} T^{8}
11D4×C2D_4\times C_2 1+7428T+8632202T2+56219857920T3+711291094304619T4+56219857920p7T5+8632202p14T6+7428p21T7+p28T8 1 + 7428 T + 8632202 T^{2} + 56219857920 T^{3} + 711291094304619 T^{4} + 56219857920 p^{7} T^{5} + 8632202 p^{14} T^{6} + 7428 p^{21} T^{7} + p^{28} T^{8}
13D4D_{4} (1+70p2T+160198410T2+70p9T3+p14T4)2 ( 1 + 70 p^{2} T + 160198410 T^{2} + 70 p^{9} T^{3} + p^{14} T^{4} )^{2}
17D4×C2D_4\times C_2 115792T16280606pT2+4651056365760T3+6130718726782755T4+4651056365760p7T516280606p15T615792p21T7+p28T8 1 - 15792 T - 16280606 p T^{2} + 4651056365760 T^{3} + 6130718726782755 T^{4} + 4651056365760 p^{7} T^{5} - 16280606 p^{15} T^{6} - 15792 p^{21} T^{7} + p^{28} T^{8}
19D4×C2D_4\times C_2 126614T1253942090T24644239023312T3+2418273122215699231T44644239023312p7T51253942090p14T626614p21T7+p28T8 1 - 26614 T - 1253942090 T^{2} - 4644239023312 T^{3} + 2418273122215699231 T^{4} - 4644239023312 p^{7} T^{5} - 1253942090 p^{14} T^{6} - 26614 p^{21} T^{7} + p^{28} T^{8}
23D4×C2D_4\times C_2 1+32640T+2057178482T2254639647088640T314236548990664871085T4254639647088640p7T5+2057178482p14T6+32640p21T7+p28T8 1 + 32640 T + 2057178482 T^{2} - 254639647088640 T^{3} - 14236548990664871085 T^{4} - 254639647088640 p^{7} T^{5} + 2057178482 p^{14} T^{6} + 32640 p^{21} T^{7} + p^{28} T^{8}
29D4D_{4} (1+158016T+39988772806T2+158016p7T3+p14T4)2 ( 1 + 158016 T + 39988772806 T^{2} + 158016 p^{7} T^{3} + p^{14} T^{4} )^{2}
31D4×C2D_4\times C_2 1+180740T5817373358T22989603578895360T3 1 + 180740 T - 5817373358 T^{2} - 2989603578895360 T^{3} - 17 ⁣ ⁣5717\!\cdots\!57T42989603578895360p7T55817373358p14T6+180740p21T7+p28T8 T^{4} - 2989603578895360 p^{7} T^{5} - 5817373358 p^{14} T^{6} + 180740 p^{21} T^{7} + p^{28} T^{8}
37D4×C2D_4\times C_2 145824T+20185374250T2+9529068243880960T3 1 - 45824 T + 20185374250 T^{2} + 9529068243880960 T^{3} - 88 ⁣ ⁣2188\!\cdots\!21T4+9529068243880960p7T5+20185374250p14T645824p21T7+p28T8 T^{4} + 9529068243880960 p^{7} T^{5} + 20185374250 p^{14} T^{6} - 45824 p^{21} T^{7} + p^{28} T^{8}
41D4D_{4} (1321720T+181439440606T2321720p7T3+p14T4)2 ( 1 - 321720 T + 181439440606 T^{2} - 321720 p^{7} T^{3} + p^{14} T^{4} )^{2}
43D4D_{4} (11023868T+671194246566T21023868p7T3+p14T4)2 ( 1 - 1023868 T + 671194246566 T^{2} - 1023868 p^{7} T^{3} + p^{14} T^{4} )^{2}
47D4×C2D_4\times C_2 11665972T+1099984870706T21103259291706623744T3+ 1 - 1665972 T + 1099984870706 T^{2} - 1103259291706623744 T^{3} + 11 ⁣ ⁣2311\!\cdots\!23T41103259291706623744p7T5+1099984870706p14T61665972p21T7+p28T8 T^{4} - 1103259291706623744 p^{7} T^{5} + 1099984870706 p^{14} T^{6} - 1665972 p^{21} T^{7} + p^{28} T^{8}
53D4×C2D_4\times C_2 1410628T2165589726190T2+6248608032034800T3+ 1 - 410628 T - 2165589726190 T^{2} + 6248608032034800 T^{3} + 38 ⁣ ⁣9938\!\cdots\!99T4+6248608032034800p7T52165589726190p14T6410628p21T7+p28T8 T^{4} + 6248608032034800 p^{7} T^{5} - 2165589726190 p^{14} T^{6} - 410628 p^{21} T^{7} + p^{28} T^{8}
59D4×C2D_4\times C_2 11702134T1128927975562T2+1618924907272816080T3+ 1 - 1702134 T - 1128927975562 T^{2} + 1618924907272816080 T^{3} + 28 ⁣ ⁣9928\!\cdots\!99T4+1618924907272816080p7T51128927975562p14T61702134p21T7+p28T8 T^{4} + 1618924907272816080 p^{7} T^{5} - 1128927975562 p^{14} T^{6} - 1702134 p^{21} T^{7} + p^{28} T^{8}
61D4×C2D_4\times C_2 1+547526T5309538579326T2370216478913573040T3+ 1 + 547526 T - 5309538579326 T^{2} - 370216478913573040 T^{3} + 20 ⁣ ⁣6720\!\cdots\!67T4370216478913573040p7T55309538579326p14T6+547526p21T7+p28T8 T^{4} - 370216478913573040 p^{7} T^{5} - 5309538579326 p^{14} T^{6} + 547526 p^{21} T^{7} + p^{28} T^{8}
67D4×C2D_4\times C_2 12590616T+2056450789210T2+19343048712604086400T3 1 - 2590616 T + 2056450789210 T^{2} + 19343048712604086400 T^{3} - 55 ⁣ ⁣0155\!\cdots\!01T4+19343048712604086400p7T5+2056450789210p14T62590616p21T7+p28T8 T^{4} + 19343048712604086400 p^{7} T^{5} + 2056450789210 p^{14} T^{6} - 2590616 p^{21} T^{7} + p^{28} T^{8}
71D4D_{4} (14129272T+22218672158062T24129272p7T3+p14T4)2 ( 1 - 4129272 T + 22218672158062 T^{2} - 4129272 p^{7} T^{3} + p^{14} T^{4} )^{2}
73D4×C2D_4\times C_2 1+8008868T+26118740970730T2+1747516196782370000pT3+ 1 + 8008868 T + 26118740970730 T^{2} + 1747516196782370000 p T^{3} + 11 ⁣ ⁣3111\!\cdots\!31p2T4+1747516196782370000p8T5+26118740970730p14T6+8008868p21T7+p28T8 p^{2} T^{4} + 1747516196782370000 p^{8} T^{5} + 26118740970730 p^{14} T^{6} + 8008868 p^{21} T^{7} + p^{28} T^{8}
79D4×C2D_4\times C_2 1+2470456T+19757905667170T2 1 + 2470456 T + 19757905667170 T^{2} - 12 ⁣ ⁣1212\!\cdots\!12T3 T^{3} - 29 ⁣ ⁣4929\!\cdots\!49T4 T^{4} - 12 ⁣ ⁣1212\!\cdots\!12p7T5+19757905667170p14T6+2470456p21T7+p28T8 p^{7} T^{5} + 19757905667170 p^{14} T^{6} + 2470456 p^{21} T^{7} + p^{28} T^{8}
83D4D_{4} (19900786T+68835957963214T29900786p7T3+p14T4)2 ( 1 - 9900786 T + 68835957963214 T^{2} - 9900786 p^{7} T^{3} + p^{14} T^{4} )^{2}
89D4×C2D_4\times C_2 115423492T+94566779700986T2 1 - 15423492 T + 94566779700986 T^{2} - 84 ⁣ ⁣4084\!\cdots\!40T3+ T^{3} + 80 ⁣ ⁣8780\!\cdots\!87T4 T^{4} - 84 ⁣ ⁣4084\!\cdots\!40p7T5+94566779700986p14T615423492p21T7+p28T8 p^{7} T^{5} + 94566779700986 p^{14} T^{6} - 15423492 p^{21} T^{7} + p^{28} T^{8}
97D4D_{4} (117377472T+164552259333822T217377472p7T3+p14T4)2 ( 1 - 17377472 T + 164552259333822 T^{2} - 17377472 p^{7} T^{3} + p^{14} T^{4} )^{2}
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   L(s)=p j=18(1αj,pps)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

−7.70104550776557944944898344050, −7.40037343241422406961754195505, −7.30902306289940021185878171534, −7.28985246771308818698677641825, −7.03503218981262130220260253290, −6.18874456559915966245969273110, −6.10365673643291093943405340781, −5.78943584648158753039347791396, −5.57364347261514177000946544966, −5.28528784141347941295761509725, −4.87053457832152162721405372955, −4.75400070048858678017954387518, −4.72504937983569954058748158577, −4.00333840680333740177532797361, −3.73243684450409317502394283715, −3.43063295030872095826499338234, −3.07598752514072215909753051672, −2.46253907578939283267205469826, −2.25810018513896790548440165901, −2.21423970157092315197588024821, −2.11830791649777830270543068506, −0.977529502770832894696557062749, −0.842447520547978161789360708025, −0.68671340316128552120876657741, −0.24913077485483568455998933811, 0.24913077485483568455998933811, 0.68671340316128552120876657741, 0.842447520547978161789360708025, 0.977529502770832894696557062749, 2.11830791649777830270543068506, 2.21423970157092315197588024821, 2.25810018513896790548440165901, 2.46253907578939283267205469826, 3.07598752514072215909753051672, 3.43063295030872095826499338234, 3.73243684450409317502394283715, 4.00333840680333740177532797361, 4.72504937983569954058748158577, 4.75400070048858678017954387518, 4.87053457832152162721405372955, 5.28528784141347941295761509725, 5.57364347261514177000946544966, 5.78943584648158753039347791396, 6.10365673643291093943405340781, 6.18874456559915966245969273110, 7.03503218981262130220260253290, 7.28985246771308818698677641825, 7.30902306289940021185878171534, 7.40037343241422406961754195505, 7.70104550776557944944898344050

Graph of the ZZ-function along the critical line