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Properties

Label 8-13e4-1.1-c2e4-0-0
Degree 88
Conductor 2856128561
Sign 11
Analytic cond. 0.01574390.0157439
Root an. cond. 0.5951670.595167
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  − 4·2-s − 4·3-s + 8·4-s + 8·5-s + 16·6-s − 12·7-s − 4·8-s − 6·9-s − 32·10-s − 4·11-s − 32·12-s + 8·13-s + 48·14-s − 32·15-s − 25·16-s + 24·18-s + 64·20-s + 48·21-s + 16·22-s + 16·24-s + 32·25-s − 32·26-s + 40·27-s − 96·28-s + 40·29-s + 128·30-s + 40·31-s + ⋯
L(s)  = 1  − 2·2-s − 4/3·3-s + 2·4-s + 8/5·5-s + 8/3·6-s − 1.71·7-s − 1/2·8-s − 2/3·9-s − 3.19·10-s − 0.363·11-s − 8/3·12-s + 8/13·13-s + 24/7·14-s − 2.13·15-s − 1.56·16-s + 4/3·18-s + 16/5·20-s + 16/7·21-s + 8/11·22-s + 2/3·24-s + 1.27·25-s − 1.23·26-s + 1.48·27-s − 3.42·28-s + 1.37·29-s + 4.26·30-s + 1.29·31-s + ⋯

Functional equation

Λ(s)=(28561s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(28561s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 2856128561    =    13413^{4}
Sign: 11
Analytic conductor: 0.01574390.0157439
Root analytic conductor: 0.5951670.595167
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28561, ( :1,1,1,1), 1)(8,\ 28561,\ (\ :1, 1, 1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.10953460180.1095346018
L(12)L(\frac12) ≈\approx 0.10953460180.1095346018
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)
bad13C22C_2^2 1−8T+8pT2−8p2T3+p4T4 1 - 8 T + 8 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4}
good2D4×C2D_4\times C_2 1+p2T+p3T2+p2T3−7T4+p4T5+p7T6+p8T7+p8T8 1 + p^{2} T + p^{3} T^{2} + p^{2} T^{3} - 7 T^{4} + p^{4} T^{5} + p^{7} T^{6} + p^{8} T^{7} + p^{8} T^{8}
3D4D_{4} (1+2T+p2T2+2p2T3+p4T4)2 ( 1 + 2 T + p^{2} T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
5D4×C2D_4\times C_2 1−8T+32T2−224T3+1559T4−224p2T5+32p4T6−8p6T7+p8T8 1 - 8 T + 32 T^{2} - 224 T^{3} + 1559 T^{4} - 224 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8}
7D4×C2D_4\times C_2 1+12T+72T2+744T3+7519T4+744p2T5+72p4T6+12p6T7+p8T8 1 + 12 T + 72 T^{2} + 744 T^{3} + 7519 T^{4} + 744 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8}
11D4×C2D_4\times C_2 1+4T+8T2+172T3−2386T4+172p2T5+8p4T6+4p6T7+p8T8 1 + 4 T + 8 T^{2} + 172 T^{3} - 2386 T^{4} + 172 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8}
17D4×C2D_4\times C_2 1−418T2+197763T4−418p4T6+p8T8 1 - 418 T^{2} + 197763 T^{4} - 418 p^{4} T^{6} + p^{8} T^{8}
19C23C_2^3 1+232162T4+p8T8 1 + 232162 T^{4} + p^{8} T^{8}
23D4×C2D_4\times C_2 1−56pT2+857778T4−56p5T6+p8T8 1 - 56 p T^{2} + 857778 T^{4} - 56 p^{5} T^{6} + p^{8} T^{8}
29D4D_{4} (1−20T+1532T2−20p2T3+p4T4)2 ( 1 - 20 T + 1532 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
31D4×C2D_4\times C_2 1−40T+800T2−39240T3+1924322T4−39240p2T5+800p4T6−40p6T7+p8T8 1 - 40 T + 800 T^{2} - 39240 T^{3} + 1924322 T^{4} - 39240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8}
37D4×C2D_4\times C_2 1−40T+800T2−46560T3+2667767T4−46560p2T5+800p4T6−40p6T7+p8T8 1 - 40 T + 800 T^{2} - 46560 T^{3} + 2667767 T^{4} - 46560 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8}
41D4×C2D_4\times C_2 1−32T+512T2−57248T3+6389378T4−57248p2T5+512p4T6−32p6T7+p8T8 1 - 32 T + 512 T^{2} - 57248 T^{3} + 6389378 T^{4} - 57248 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8}
43D4×C2D_4\times C_2 1−6334T2+16708731T4−6334p4T6+p8T8 1 - 6334 T^{2} + 16708731 T^{4} - 6334 p^{4} T^{6} + p^{8} T^{8}
47D4×C2D_4\times C_2 1+4T+8T2−1736T3−6608737T4−1736p2T5+8p4T6+4p6T7+p8T8 1 + 4 T + 8 T^{2} - 1736 T^{3} - 6608737 T^{4} - 1736 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8}
53D4D_{4} (1+40T+5768T2+40p2T3+p4T4)2 ( 1 + 40 T + 5768 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2}
59D4×C2D_4\times C_2 1−56T+1568T2+2632T3−12442366T4+2632p2T5+1568p4T6−56p6T7+p8T8 1 - 56 T + 1568 T^{2} + 2632 T^{3} - 12442366 T^{4} + 2632 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8}
61D4D_{4} (1+148T+12878T2+148p2T3+p4T4)2 ( 1 + 148 T + 12878 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2}
67D4×C2D_4\times C_2 1+84T+3528T2−155316T3−33332642T4−155316p2T5+3528p4T6+84p6T7+p8T8 1 + 84 T + 3528 T^{2} - 155316 T^{3} - 33332642 T^{4} - 155316 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8}
71D4×C2D_4\times C_2 1−4pT+8p2T2−57112pT3+322400399T4−57112p3T5+8p6T6−4p7T7+p8T8 1 - 4 p T + 8 p^{2} T^{2} - 57112 p T^{3} + 322400399 T^{4} - 57112 p^{3} T^{5} + 8 p^{6} T^{6} - 4 p^{7} T^{7} + p^{8} T^{8}
73C23C_2^3 1+18164482T4+p8T8 1 + 18164482 T^{4} + p^{8} T^{8}
79D4D_{4} (1−32T+11528T2−32p2T3+p4T4)2 ( 1 - 32 T + 11528 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4×C2D_4\times C_2 1+52T+1352T2+271804T3+51880814T4+271804p2T5+1352p4T6+52p6T7+p8T8 1 + 52 T + 1352 T^{2} + 271804 T^{3} + 51880814 T^{4} + 271804 p^{2} T^{5} + 1352 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−200T+20000T2−1908200T3+179436962T4−1908200p2T5+20000p4T6−200p6T7+p8T8 1 - 200 T + 20000 T^{2} - 1908200 T^{3} + 179436962 T^{4} - 1908200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8}
97D4×C2D_4\times C_2 1+68T+2312T2−801924T3−171375106T4−801924p2T5+2312p4T6+68p6T7+p8T8 1 + 68 T + 2312 T^{2} - 801924 T^{3} - 171375106 T^{4} - 801924 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8}
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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

−15.73267954907706978465117940994, −14.90831937210881891886498767921, −14.45109489363855657383888504907, −13.69557914963411258049827622329, −13.67826964445310405211522983516, −13.56268633963231116245228884102, −12.73477724760375199072324025869, −12.48767843173433937554344724520, −12.03463667937509193806138129804, −11.43954033197981425354098322200, −10.99395777799112031275258853299, −10.65636187705288183345272844949, −10.50876836275333162196408802027, −9.896974242426991948391952809843, −9.563996724071608323299871578343, −9.177307123611610263205322290920, −8.998398593171744386351339134601, −8.056957803555827474452859535269, −7.921204180954394322383080744709, −6.85874272103586218025205582972, −6.38261627920717730506117029383, −6.16404177135625374340439059988, −5.65990377670370487912622172668, −4.77450417013775926355845063463, −2.85218477777242428197479268848, 2.85218477777242428197479268848, 4.77450417013775926355845063463, 5.65990377670370487912622172668, 6.16404177135625374340439059988, 6.38261627920717730506117029383, 6.85874272103586218025205582972, 7.921204180954394322383080744709, 8.056957803555827474452859535269, 8.998398593171744386351339134601, 9.177307123611610263205322290920, 9.563996724071608323299871578343, 9.896974242426991948391952809843, 10.50876836275333162196408802027, 10.65636187705288183345272844949, 10.99395777799112031275258853299, 11.43954033197981425354098322200, 12.03463667937509193806138129804, 12.48767843173433937554344724520, 12.73477724760375199072324025869, 13.56268633963231116245228884102, 13.67826964445310405211522983516, 13.69557914963411258049827622329, 14.45109489363855657383888504907, 14.90831937210881891886498767921, 15.73267954907706978465117940994

Graph of the ZZ-function along the critical line