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Properties

Label 4-43904-1.1-c1e2-0-9
Degree 44
Conductor 4390443904
Sign 11
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 5·9-s − 14-s + 16-s + 3·17-s + 5·18-s − 3·23-s − 2·25-s − 28-s − 6·31-s + 32-s + 3·34-s + 5·36-s + 9·41-s − 3·46-s − 15·47-s + 49-s − 2·50-s − 56-s − 6·62-s − 5·63-s + 64-s + 3·68-s + 15·71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 5/3·9-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.17·18-s − 0.625·23-s − 2/5·25-s − 0.188·28-s − 1.07·31-s + 0.176·32-s + 0.514·34-s + 5/6·36-s + 1.40·41-s − 0.442·46-s − 2.18·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.762·62-s − 0.629·63-s + 1/8·64-s + 0.363·68-s + 1.78·71-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: 11
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 43904, ( :1/2,1/2), 1)(4,\ 43904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2136059892.213605989
L(12)L(\frac12) ≈\approx 2.2136059892.213605989
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)
bad2C1C_1 1−T 1 - T
7C1C_1 1+T 1 + T
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
13C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−6T+pT2)(1+3T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−3T+pT2)(1+6T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C22C_2^2 1+43T2+p2T4 1 + 43 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+T+pT2)(1+5T+pT2) ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} )
37C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−9T+pT2)(1+pT2) ( 1 - 9 T + p T^{2} )( 1 + p T^{2} )
43C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1+3T+pT2)(1+12T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C22C_2^2 1+85T2+p2T4 1 + 85 T^{2} + p^{2} T^{4}
59C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
61C22C_2^2 1−40T2+p2T4 1 - 40 T^{2} + p^{2} T^{4}
67C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−9T+pT2)(1−6T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
83C22C_2^2 1+5T2+p2T4 1 + 5 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+3T+pT2)(1+12T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.05678375381477042318874611224, −9.756948225725246098638511465202, −9.502508101509796184785562727000, −8.572471043036661544792716627202, −7.959398509703870639269961294849, −7.40680873284187474787785819703, −7.08975345681600839738665962505, −6.35386202430917741466320291891, −5.93928509818591661972668925621, −5.13656851188741939169424055778, −4.60634381509827538369531600859, −3.84629395034157933487320961776, −3.48844498554738898579609102006, −2.34612635281934047519280732349, −1.42882639006212863066694207280, 1.42882639006212863066694207280, 2.34612635281934047519280732349, 3.48844498554738898579609102006, 3.84629395034157933487320961776, 4.60634381509827538369531600859, 5.13656851188741939169424055778, 5.93928509818591661972668925621, 6.35386202430917741466320291891, 7.08975345681600839738665962505, 7.40680873284187474787785819703, 7.959398509703870639269961294849, 8.572471043036661544792716627202, 9.502508101509796184785562727000, 9.756948225725246098638511465202, 10.05678375381477042318874611224

Graph of the ZZ-function along the critical line