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Properties

Label 2-346560-1.1-c1-0-10
Degree 22
Conductor 346560346560
Sign 11
Analytic cond. 2767.292767.29
Root an. cond. 52.605052.6050
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  − 3-s − 5-s − 2·7-s + 9-s + 2·11-s + 15-s − 2·17-s + 2·21-s − 8·23-s + 25-s − 27-s − 2·33-s + 2·35-s + 4·37-s + 8·41-s + 6·43-s − 45-s − 8·47-s − 3·49-s + 2·51-s − 10·53-s − 2·55-s − 8·59-s − 2·61-s − 2·63-s + 8·69-s − 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 0.485·17-s + 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.348·33-s + 0.338·35-s + 0.657·37-s + 1.24·41-s + 0.914·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 0.269·55-s − 1.04·59-s − 0.256·61-s − 0.251·63-s + 0.963·69-s − 0.949·71-s + ⋯

Functional equation

Λ(s)=(346560s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(346560s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 346560346560    =    26⋅3⋅5⋅1922^{6} \cdot 3 \cdot 5 \cdot 19^{2}
Sign: 11
Analytic conductor: 2767.292767.29
Root analytic conductor: 52.605052.6050
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 346560, ( :1/2), 1)(2,\ 346560,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.34692684370.3469268437
L(12)L(\frac12) ≈\approx 0.34692684370.3469268437
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}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
5 1+T 1 + T
19 1 1
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1+8T+pT2 1 + 8 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.45977793783298, −12.19379299696375, −11.62084594829255, −11.23549275730637, −10.84083743768355, −10.33480723150869, −9.774899107414703, −9.395649616557565, −9.120401433693890, −8.319675015894902, −7.917227482294149, −7.538241371417994, −6.865697583072545, −6.472211173002201, −6.031409698469904, −5.780020244770819, −4.908599873114431, −4.454307724081260, −4.078164510075542, −3.518308233738156, −2.982644229903113, −2.314166151976587, −1.663519367388139, −1.015041681683283, −0.1780199181726390, 0.1780199181726390, 1.015041681683283, 1.663519367388139, 2.314166151976587, 2.982644229903113, 3.518308233738156, 4.078164510075542, 4.454307724081260, 4.908599873114431, 5.780020244770819, 6.031409698469904, 6.472211173002201, 6.865697583072545, 7.538241371417994, 7.917227482294149, 8.319675015894902, 9.120401433693890, 9.395649616557565, 9.774899107414703, 10.33480723150869, 10.84083743768355, 11.23549275730637, 11.62084594829255, 12.19379299696375, 12.45977793783298

Graph of the ZZ-function along the critical line