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Properties

Label 8-370e4-1.1-c1e4-0-11
Degree 88
Conductor 1874161000018741610000
Sign 11
Analytic cond. 76.193076.1930
Root an. cond. 1.718851.71885
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4-s − 6·5-s + 6·6-s − 12·7-s + 2·8-s + 2·9-s + 12·10-s − 3·12-s − 5·13-s + 24·14-s + 18·15-s − 4·16-s − 3·17-s − 4·18-s − 6·20-s + 36·21-s − 12·23-s − 6·24-s + 17·25-s + 10·26-s + 3·27-s − 12·28-s − 36·30-s + 2·32-s + 6·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.68·5-s + 2.44·6-s − 4.53·7-s + 0.707·8-s + 2/3·9-s + 3.79·10-s − 0.866·12-s − 1.38·13-s + 6.41·14-s + 4.64·15-s − 16-s − 0.727·17-s − 0.942·18-s − 1.34·20-s + 7.85·21-s − 2.50·23-s − 1.22·24-s + 17/5·25-s + 1.96·26-s + 0.577·27-s − 2.26·28-s − 6.57·30-s + 0.353·32-s + 1.02·34-s + ⋯

Functional equation

Λ(s)=((24⋅54⋅374)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅54⋅374)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅54⋅3742^{4} \cdot 5^{4} \cdot 37^{4}
Sign: 11
Analytic conductor: 76.193076.1930
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 24⋅54⋅374, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T+T2)2 ( 1 + T + T^{2} )^{2}
5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
37C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
good3C2C_2×\timesC22C_2^2 (1+T+pT2)2(1+T−2T2+pT3+p2T4) ( 1 + T + p T^{2} )^{2}( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )
7C2C_2 (1+T+pT2)2(1+5T+pT2)2 ( 1 + T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2}
11C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
13D4×C2D_4\times C_2 1+5T+T2−10T3+82T4−10pT5+p2T6+5p3T7+p4T8 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
17D4×C2D_4\times C_2 1+3T−19T2−18T3+342T4−18pT5−19p2T6+3p3T7+p4T8 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
19C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
23D4D_{4} (1+6T+22T2+6pT3+p2T4)2 ( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
29D4×C2D_4\times C_2 1−37T2+1620T4−37p2T6+p4T8 1 - 37 T^{2} + 1620 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−T2−1716T4−p2T6+p4T8 1 - T^{2} - 1716 T^{4} - p^{2} T^{6} + p^{4} T^{8}
41C23C_2^3 1−49T2+720T4−49p2T6+p4T8 1 - 49 T^{2} + 720 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8}
43D4D_{4} (1+T+12T2+pT3+p2T4)2 ( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
47D4×C2D_4\times C_2 1−160T2+10686T4−160p2T6+p4T8 1 - 160 T^{2} + 10686 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−9T+137T2−990T3+10722T4−990pT5+137p2T6−9p3T7+p4T8 1 - 9 T + 137 T^{2} - 990 T^{3} + 10722 T^{4} - 990 p T^{5} + 137 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
59D4×C2D_4\times C_2 1+30T+482T2+5460T3+47343T4+5460pT5+482p2T6+30p3T7+p4T8 1 + 30 T + 482 T^{2} + 5460 T^{3} + 47343 T^{4} + 5460 p T^{5} + 482 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+9T+131T2+936T3+8742T4+936pT5+131p2T6+9p3T7+p4T8 1 + 9 T + 131 T^{2} + 936 T^{3} + 8742 T^{4} + 936 p T^{5} + 131 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−6T+50T2−228T3−2241T4−228pT5+50p2T6−6p3T7+p4T8 1 - 6 T + 50 T^{2} - 228 T^{3} - 2241 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−6T−82T2+144T3+6327T4+144pT5−82p2T6−6p3T7+p4T8 1 - 6 T - 82 T^{2} + 144 T^{3} + 6327 T^{4} + 144 p T^{5} - 82 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
73C22C_2^2 (1−98T2+p2T4)2 ( 1 - 98 T^{2} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−6T+74T2−372T3−1449T4−372pT5+74p2T6−6p3T7+p4T8 1 - 6 T + 74 T^{2} - 372 T^{3} - 1449 T^{4} - 372 p T^{5} + 74 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1+24T+362T2+4080T3+37947T4+4080pT5+362p2T6+24p3T7+p4T8 1 + 24 T + 362 T^{2} + 4080 T^{3} + 37947 T^{4} + 4080 p T^{5} + 362 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
89D4×C2D_4\times C_2 1+39T+809T2+11778T3+128406T4+11778pT5+809p2T6+39p3T7+p4T8 1 + 39 T + 809 T^{2} + 11778 T^{3} + 128406 T^{4} + 11778 p T^{5} + 809 p^{2} T^{6} + 39 p^{3} T^{7} + p^{4} T^{8}
97D4D_{4} (1+7T+132T2+7pT3+p2T4)2 ( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} 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

−8.894512461875042411245339024924, −8.495237136544163940562244010178, −8.201838672917920221575809326230, −8.124948522730085121833255073320, −7.68156597875810815462165669009, −7.54642146519005609743315884400, −7.14638913893938203619892257692, −7.08054401987464178557179478376, −6.89633244033457933420439212749, −6.43710549241103880379807921398, −6.39664827871028382805597708233, −6.11740602243252145076170784437, −5.82907691661979055338889169674, −5.60981092847855310671001524995, −5.34272215297827295424750096398, −4.59102875602102777040626949036, −4.46377013943940548994635233799, −4.35166462100043483663066563548, −3.90912450530183781296641046838, −3.55242703230157424368167477410, −3.36293840380481594562593739434, −3.12616451694725125542948197612, −2.55310379476121640911379277657, −2.40446390556957536163841765474, −1.19940740588260723906817380778, 0, 0, 0, 0, 1.19940740588260723906817380778, 2.40446390556957536163841765474, 2.55310379476121640911379277657, 3.12616451694725125542948197612, 3.36293840380481594562593739434, 3.55242703230157424368167477410, 3.90912450530183781296641046838, 4.35166462100043483663066563548, 4.46377013943940548994635233799, 4.59102875602102777040626949036, 5.34272215297827295424750096398, 5.60981092847855310671001524995, 5.82907691661979055338889169674, 6.11740602243252145076170784437, 6.39664827871028382805597708233, 6.43710549241103880379807921398, 6.89633244033457933420439212749, 7.08054401987464178557179478376, 7.14638913893938203619892257692, 7.54642146519005609743315884400, 7.68156597875810815462165669009, 8.124948522730085121833255073320, 8.201838672917920221575809326230, 8.495237136544163940562244010178, 8.894512461875042411245339024924

Graph of the ZZ-function along the critical line