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Properties

Label 2-3120-1.1-c1-0-6
Degree 22
Conductor 31203120
Sign 11
Analytic cond. 24.913324.9133
Root an. cond. 4.991324.99132
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 + 13-s − 15-s − 2·19-s + 2·21-s + 6·23-s + 25-s − 27-s − 8·31-s − 2·35-s + 2·37-s − 39-s + 6·41-s + 4·43-s + 45-s − 3·49-s − 6·53-s + 2·57-s + 14·61-s − 2·63-s + 65-s + 4·67-s − 6·69-s − 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 3/7·49-s − 0.824·53-s + 0.264·57-s + 1.79·61-s − 0.251·63-s + 0.124·65-s + 0.488·67-s − 0.722·69-s − 0.468·73-s + ⋯

Functional equation

Λ(s)=(3120s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3120s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31203120    =    24⋅3⋅5⋅132^{4} \cdot 3 \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 24.913324.9133
Root analytic conductor: 4.991324.99132
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3120, ( :1/2), 1)(2,\ 3120,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3919585331.391958533
L(12)L(\frac12) ≈\approx 1.3919585331.391958533
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
13 1−T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+4T+pT2 1 + 4 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

−8.907998718246193749300650634254, −7.85832220713197698190585116152, −6.99686575372866242821342185369, −6.42262701034001882837697519636, −5.70233605244569122682417518238, −4.98674800015260782294546923298, −3.98803889555059043901076634706, −3.09638465664731709775430870739, −2.00888213860569869943119252787, −0.73199284761332162352741133507, 0.73199284761332162352741133507, 2.00888213860569869943119252787, 3.09638465664731709775430870739, 3.98803889555059043901076634706, 4.98674800015260782294546923298, 5.70233605244569122682417518238, 6.42262701034001882837697519636, 6.99686575372866242821342185369, 7.85832220713197698190585116152, 8.907998718246193749300650634254

Graph of the ZZ-function along the critical line