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Properties

Label 4-43904-1.1-c1e2-0-14
Degree 44
Conductor 4390443904
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s − 3·17-s + 18-s − 15·23-s − 2·25-s − 28-s − 12·31-s − 32-s + 3·34-s − 36-s − 3·41-s + 15·46-s + 9·47-s + 49-s + 2·50-s + 56-s + 12·62-s + 63-s + 64-s − 3·68-s + 9·71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1/3·9-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 3.12·23-s − 2/5·25-s − 0.188·28-s − 2.15·31-s − 0.176·32-s + 0.514·34-s − 1/6·36-s − 0.468·41-s + 2.21·46-s + 1.31·47-s + 1/7·49-s + 0.282·50-s + 0.133·56-s + 1.52·62-s + 0.125·63-s + 1/8·64-s − 0.363·68-s + 1.06·71-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: −1-1
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: 11
Selberg data: (4, 43904, ( :1/2,1/2), −1)(4,\ 43904,\ (\ :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)
bad2C1C_1 1+T 1 + T
7C1C_1 1+T 1 + T
good3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−3T+pT2)(1+6T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+6T+pT2)(1+9T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} )
29C22C_2^2 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+5T+pT2)(1+7T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C22C_2^2 1+7T2+p2T4 1 + 7 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1+pT2)(1+3T+pT2) ( 1 + p T^{2} )( 1 + 3 T + p T^{2} )
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−12T+pT2)(1+3T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} )
53C22C_2^2 1−95T2+p2T4 1 - 95 T^{2} + p^{2} T^{4}
59C22C_2^2 1+101T2+p2T4 1 + 101 T^{2} + p^{2} T^{4}
61C22C_2^2 1−40T2+p2T4 1 - 40 T^{2} + p^{2} T^{4}
67C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−6T+pT2)(1−3T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−16T+pT2)(1−2T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
83C22C_2^2 1−145T2+p2T4 1 - 145 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+pT2)(1+9T+pT2) ( 1 + p T^{2} )( 1 + 9 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−14T+pT2)(1−10T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 10 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

−9.903259475993730301133601247932, −9.400946509192316215215642760736, −9.032872625967611552960835581299, −8.322439564034960678023719361421, −7.986082184770432412320687121706, −7.40607211260737083787304764859, −6.80999099230277597284059767890, −6.15879308148576935335469093848, −5.78902777848912703504487358241, −5.08639353379536435232603331513, −3.92316526135323766397833934784, −3.75383713546245393225954256150, −2.46401242958316858193289692726, −1.88856445010374489988253513893, 0, 1.88856445010374489988253513893, 2.46401242958316858193289692726, 3.75383713546245393225954256150, 3.92316526135323766397833934784, 5.08639353379536435232603331513, 5.78902777848912703504487358241, 6.15879308148576935335469093848, 6.80999099230277597284059767890, 7.40607211260737083787304764859, 7.986082184770432412320687121706, 8.322439564034960678023719361421, 9.032872625967611552960835581299, 9.400946509192316215215642760736, 9.903259475993730301133601247932

Graph of the ZZ-function along the critical line