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Properties

Label 2-3042-1.1-c1-0-15
Degree 22
Conductor 30423042
Sign 11
Analytic cond. 24.290424.2904
Root an. cond. 4.928534.92853
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.73·5-s + 1.26·7-s + 8-s − 1.73·10-s − 1.26·11-s + 1.26·14-s + 16-s − 5.19·17-s + 4.73·19-s − 1.73·20-s − 1.26·22-s + 8.19·23-s − 2.00·25-s + 1.26·28-s + 3·29-s + 9.46·31-s + 32-s − 5.19·34-s − 2.19·35-s + 3·37-s + 4.73·38-s − 1.73·40-s − 6.46·41-s − 4.19·43-s − 1.26·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.774·5-s + 0.479·7-s + 0.353·8-s − 0.547·10-s − 0.382·11-s + 0.338·14-s + 0.250·16-s − 1.26·17-s + 1.08·19-s − 0.387·20-s − 0.270·22-s + 1.70·23-s − 0.400·25-s + 0.239·28-s + 0.557·29-s + 1.69·31-s + 0.176·32-s − 0.891·34-s − 0.371·35-s + 0.493·37-s + 0.767·38-s − 0.273·40-s − 1.00·41-s − 0.639·43-s − 0.191·44-s + ⋯

Functional equation

Λ(s)=(3042s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3042s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30423042    =    2⋅32⋅1322 \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 24.290424.2904
Root analytic conductor: 4.928534.92853
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3042, ( :1/2), 1)(2,\ 3042,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6598211512.659821151
L(12)L(\frac12) ≈\approx 2.6598211512.659821151
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−T 1 - T
3 1 1
13 1 1
good5 1+1.73T+5T2 1 + 1.73T + 5T^{2}
7 1−1.26T+7T2 1 - 1.26T + 7T^{2}
11 1+1.26T+11T2 1 + 1.26T + 11T^{2}
17 1+5.19T+17T2 1 + 5.19T + 17T^{2}
19 1−4.73T+19T2 1 - 4.73T + 19T^{2}
23 1−8.19T+23T2 1 - 8.19T + 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1−9.46T+31T2 1 - 9.46T + 31T^{2}
37 1−3T+37T2 1 - 3T + 37T^{2}
41 1+6.46T+41T2 1 + 6.46T + 41T^{2}
43 1+4.19T+43T2 1 + 4.19T + 43T^{2}
47 1−4.73T+47T2 1 - 4.73T + 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1−13.8T+59T2 1 - 13.8T + 59T^{2}
61 1−15.1T+61T2 1 - 15.1T + 61T^{2}
67 1−7.26T+67T2 1 - 7.26T + 67T^{2}
71 1−2.19T+71T2 1 - 2.19T + 71T^{2}
73 1−12.1T+73T2 1 - 12.1T + 73T^{2}
79 1−8.39T+79T2 1 - 8.39T + 79T^{2}
83 1+5.66T+83T2 1 + 5.66T + 83T^{2}
89 1+9.46T+89T2 1 + 9.46T + 89T^{2}
97 1−6T+97T2 1 - 6T + 97T^{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

−8.394954276680925924230407291964, −8.054524933309666890774180561198, −6.99733113537378081800058731230, −6.64322499263362940144347190670, −5.36069361030688311095974606106, −4.87090390824712768481786189131, −4.08189789689010512488697510855, −3.17280689697497580960092811354, −2.31979612613075247521586249959, −0.909448153655508590737341082519, 0.909448153655508590737341082519, 2.31979612613075247521586249959, 3.17280689697497580960092811354, 4.08189789689010512488697510855, 4.87090390824712768481786189131, 5.36069361030688311095974606106, 6.64322499263362940144347190670, 6.99733113537378081800058731230, 8.054524933309666890774180561198, 8.394954276680925924230407291964

Graph of the ZZ-function along the critical line