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Properties

Label 8-132e4-1.1-c2e4-0-1
Degree 88
Conductor 303595776303595776
Sign 11
Analytic cond. 167.353167.353
Root an. cond. 1.896501.89650
Motivic weight 22
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  − 4·3-s − 6·9-s + 48·13-s − 56·19-s − 4·25-s + 76·27-s + 24·31-s + 120·37-s − 192·39-s − 120·43-s − 20·49-s + 224·57-s − 48·61-s + 136·67-s − 104·73-s + 16·75-s − 192·79-s − 109·81-s − 96·93-s − 360·97-s + 168·103-s + 128·109-s − 480·111-s − 288·117-s − 22·121-s + 127-s + 480·129-s + ⋯
L(s)  = 1  − 4/3·3-s − 2/3·9-s + 3.69·13-s − 2.94·19-s − 0.159·25-s + 2.81·27-s + 0.774·31-s + 3.24·37-s − 4.92·39-s − 2.79·43-s − 0.408·49-s + 3.92·57-s − 0.786·61-s + 2.02·67-s − 1.42·73-s + 0.213·75-s − 2.43·79-s − 1.34·81-s − 1.03·93-s − 3.71·97-s + 1.63·103-s + 1.17·109-s − 4.32·111-s − 2.46·117-s − 0.181·121-s + 0.00787·127-s + 3.72·129-s + ⋯

Functional equation

Λ(s)=((28⋅34⋅114)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((28⋅34⋅114)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅34⋅1142^{8} \cdot 3^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 167.353167.353
Root analytic conductor: 1.896501.89650
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅34⋅114, ( :1,1,1,1), 1)(8,\ 2^{8} \cdot 3^{4} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.96778166480.9677816648
L(12)L(\frac12) ≈\approx 0.96778166480.9677816648
L(2)L(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)
bad2 1 1
3C2C_2 (1+2T+p2T2)2 ( 1 + 2 T + p^{2} T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
good5D4×C2D_4\times C_2 1+4T2−154T4+4p4T6+p8T8 1 + 4 T^{2} - 154 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8}
7C22C_2^2 (1+10T2+p4T4)2 ( 1 + 10 T^{2} + p^{4} T^{4} )^{2}
13D4D_{4} (1−24T+394T2−24p2T3+p4T4)2 ( 1 - 24 T + 394 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2}
17D4×C2D_4\times C_2 1−548T2+152006T4−548p4T6+p8T8 1 - 548 T^{2} + 152006 T^{4} - 548 p^{4} T^{6} + p^{8} T^{8}
19D4D_{4} (1+28T+566T2+28p2T3+p4T4)2 ( 1 + 28 T + 566 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4×C2D_4\times C_2 1−1180T2+793734T4−1180p4T6+p8T8 1 - 1180 T^{2} + 793734 T^{4} - 1180 p^{4} T^{6} + p^{8} T^{8}
29D4×C2D_4\times C_2 1−2948T2+3564710T4−2948p4T6+p8T8 1 - 2948 T^{2} + 3564710 T^{4} - 2948 p^{4} T^{6} + p^{8} T^{8}
31D4D_{4} (1−12T+1606T2−12p2T3+p4T4)2 ( 1 - 12 T + 1606 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2}
37D4D_{4} (1−60T+3286T2−60p2T3+p4T4)2 ( 1 - 60 T + 3286 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4×C2D_4\times C_2 1−3236T2+7165574T4−3236p4T6+p8T8 1 - 3236 T^{2} + 7165574 T^{4} - 3236 p^{4} T^{6} + p^{8} T^{8}
43D4D_{4} (1+60T+4246T2+60p2T3+p4T4)2 ( 1 + 60 T + 4246 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4×C2D_4\times C_2 1−7964T2+25546694T4−7964p4T6+p8T8 1 - 7964 T^{2} + 25546694 T^{4} - 7964 p^{4} T^{6} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−6524T2+24696806T4−6524p4T6+p8T8 1 - 6524 T^{2} + 24696806 T^{4} - 6524 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1−1540T2−4368666T4−1540p4T6+p8T8 1 - 1540 T^{2} - 4368666 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+24T+7498T2+24p2T3+p4T4)2 ( 1 + 24 T + 7498 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2}
67D4D_{4} (1−68T+4502T2−68p2T3+p4T4)2 ( 1 - 68 T + 4502 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2}
71D4×C2D_4\times C_2 1−8732T2+61537286T4−8732p4T6+p8T8 1 - 8732 T^{2} + 61537286 T^{4} - 8732 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+52T+2534T2+52p2T3+p4T4)2 ( 1 + 52 T + 2534 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4D_{4} (1+96T+14698T2+96p2T3+p4T4)2 ( 1 + 96 T + 14698 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4×C2D_4\times C_2 1+6652T2+94061606T4+6652p4T6+p8T8 1 + 6652 T^{2} + 94061606 T^{4} + 6652 p^{4} T^{6} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−29732T2+345919238T4−29732p4T6+p8T8 1 - 29732 T^{2} + 345919238 T^{4} - 29732 p^{4} T^{6} + p^{8} T^{8}
97D4D_{4} (1+180T+25510T2+180p2T3+p4T4)2 ( 1 + 180 T + 25510 T^{2} + 180 p^{2} T^{3} + p^{4} 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

−9.577990669632024711119150319558, −9.314897175201610425668146588946, −8.691942293754224939086056408306, −8.515631402310660871799254634134, −8.409367642329261697807057404175, −8.318785141006575132198691118353, −8.166025228394766077126418199134, −7.56246846318674985110626659429, −6.92152331138720457126023628021, −6.62073772941342379783827933792, −6.58746093998387031164345176793, −6.14367602458893349767216016208, −5.95989970095986457000561774967, −5.86172324499724082875395473454, −5.59786928759725336060040295557, −4.89118291830986800707641256653, −4.64035691952586383280035149383, −4.12597517193238478663924325190, −4.11139251941347730670543369472, −3.43004255918660738661113053838, −3.04744421348196799514269874466, −2.60500773002384297956998016340, −1.80845641971328716610752250374, −1.23295726433643618389422352550, −0.46764483837531875041946054064, 0.46764483837531875041946054064, 1.23295726433643618389422352550, 1.80845641971328716610752250374, 2.60500773002384297956998016340, 3.04744421348196799514269874466, 3.43004255918660738661113053838, 4.11139251941347730670543369472, 4.12597517193238478663924325190, 4.64035691952586383280035149383, 4.89118291830986800707641256653, 5.59786928759725336060040295557, 5.86172324499724082875395473454, 5.95989970095986457000561774967, 6.14367602458893349767216016208, 6.58746093998387031164345176793, 6.62073772941342379783827933792, 6.92152331138720457126023628021, 7.56246846318674985110626659429, 8.166025228394766077126418199134, 8.318785141006575132198691118353, 8.409367642329261697807057404175, 8.515631402310660871799254634134, 8.691942293754224939086056408306, 9.314897175201610425668146588946, 9.577990669632024711119150319558

Graph of the ZZ-function along the critical line