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Properties

Label 4-43904-1.1-c1e2-0-3
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 + 2·9-s − 14-s + 16-s − 2·17-s − 2·18-s + 4·23-s + 2·25-s + 28-s + 4·31-s − 32-s + 2·34-s + 2·36-s − 2·41-s − 4·46-s + 12·47-s + 49-s − 2·50-s − 56-s − 4·62-s + 2·63-s + 64-s − 2·68-s + 20·71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 2/3·9-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.834·23-s + 2/5·25-s + 0.188·28-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 1/3·36-s − 0.312·41-s − 0.589·46-s + 1.75·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.508·62-s + 0.251·63-s + 1/8·64-s − 0.242·68-s + 2.37·71-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 1.0664217461.066421746
L(12)L(\frac12) ≈\approx 1.0664217461.066421746
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 1−2T2+p2T4 1 - 2 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 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
37C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1+pT2)(1+2T+pT2) ( 1 + p T^{2} )( 1 + 2 T + p T^{2} )
43C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )
53C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
59C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
61C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
67C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−8T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 8 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 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+86T2+p2T4 1 + 86 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−18T+pT2)(1+16T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 16 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+12T+pT2)(1+14T+pT2) ( 1 + 12 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

−10.19274505607597439649156228478, −9.677108046192235870246199356106, −9.194313781989916405496105062762, −8.528800373646140821271103303358, −8.386038696683392022429998649209, −7.39177891759976496316020420880, −7.28066803334605450056550690805, −6.60343904133397295855360218542, −6.00252687609791041703529005071, −5.25027462301996463773786138541, −4.61343572082246267383198569105, −3.96305578083704870704442222086, −3.00325091538984583711994775428, −2.18055419937459732379532815042, −1.10715236233253586523024540681, 1.10715236233253586523024540681, 2.18055419937459732379532815042, 3.00325091538984583711994775428, 3.96305578083704870704442222086, 4.61343572082246267383198569105, 5.25027462301996463773786138541, 6.00252687609791041703529005071, 6.60343904133397295855360218542, 7.28066803334605450056550690805, 7.39177891759976496316020420880, 8.386038696683392022429998649209, 8.528800373646140821271103303358, 9.194313781989916405496105062762, 9.677108046192235870246199356106, 10.19274505607597439649156228478

Graph of the ZZ-function along the critical line