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Properties

Label 2-624-1.1-c5-0-23
Degree 22
Conductor 624624
Sign 11
Analytic cond. 100.079100.079
Root an. cond. 10.003910.0039
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 82.3·5-s − 101.·7-s + 81·9-s − 154.·11-s + 169·13-s + 741.·15-s − 2.21e3·17-s − 171.·19-s − 910.·21-s + 2.49e3·23-s + 3.65e3·25-s + 729·27-s + 5.53e3·29-s + 7.17e3·31-s − 1.39e3·33-s − 8.32e3·35-s − 580.·37-s + 1.52e3·39-s + 1.18e4·41-s + 6.59e3·43-s + 6.67e3·45-s − 6.91e3·47-s − 6.58e3·49-s − 1.99e4·51-s + 3.51e4·53-s − 1.27e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.47·5-s − 0.779·7-s + 0.333·9-s − 0.385·11-s + 0.277·13-s + 0.850·15-s − 1.85·17-s − 0.108·19-s − 0.450·21-s + 0.982·23-s + 1.16·25-s + 0.192·27-s + 1.22·29-s + 1.34·31-s − 0.222·33-s − 1.14·35-s − 0.0697·37-s + 0.160·39-s + 1.10·41-s + 0.544·43-s + 0.491·45-s − 0.456·47-s − 0.391·49-s − 1.07·51-s + 1.72·53-s − 0.567·55-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 100.079100.079
Root analytic conductor: 10.003910.0039
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 624, ( :5/2), 1)(2,\ 624,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.4035765443.403576544
L(12)L(\frac12) ≈\approx 3.4035765443.403576544
L(72)L(\frac{7}{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 1−9T 1 - 9T
13 1−169T 1 - 169T
good5 1−82.3T+3.12e3T2 1 - 82.3T + 3.12e3T^{2}
7 1+101.T+1.68e4T2 1 + 101.T + 1.68e4T^{2}
11 1+154.T+1.61e5T2 1 + 154.T + 1.61e5T^{2}
17 1+2.21e3T+1.41e6T2 1 + 2.21e3T + 1.41e6T^{2}
19 1+171.T+2.47e6T2 1 + 171.T + 2.47e6T^{2}
23 1−2.49e3T+6.43e6T2 1 - 2.49e3T + 6.43e6T^{2}
29 1−5.53e3T+2.05e7T2 1 - 5.53e3T + 2.05e7T^{2}
31 1−7.17e3T+2.86e7T2 1 - 7.17e3T + 2.86e7T^{2}
37 1+580.T+6.93e7T2 1 + 580.T + 6.93e7T^{2}
41 1−1.18e4T+1.15e8T2 1 - 1.18e4T + 1.15e8T^{2}
43 1−6.59e3T+1.47e8T2 1 - 6.59e3T + 1.47e8T^{2}
47 1+6.91e3T+2.29e8T2 1 + 6.91e3T + 2.29e8T^{2}
53 1−3.51e4T+4.18e8T2 1 - 3.51e4T + 4.18e8T^{2}
59 1−2.61e4T+7.14e8T2 1 - 2.61e4T + 7.14e8T^{2}
61 1−4.28e4T+8.44e8T2 1 - 4.28e4T + 8.44e8T^{2}
67 1−5.14e4T+1.35e9T2 1 - 5.14e4T + 1.35e9T^{2}
71 1−4.66e4T+1.80e9T2 1 - 4.66e4T + 1.80e9T^{2}
73 1+1.90e4T+2.07e9T2 1 + 1.90e4T + 2.07e9T^{2}
79 1+3.74e4T+3.07e9T2 1 + 3.74e4T + 3.07e9T^{2}
83 1+1.01e5T+3.93e9T2 1 + 1.01e5T + 3.93e9T^{2}
89 1−5.95e4T+5.58e9T2 1 - 5.95e4T + 5.58e9T^{2}
97 1+8.66e4T+8.58e9T2 1 + 8.66e4T + 8.58e9T^{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

−9.805353329081282244670300954756, −9.021408372978754718610141754146, −8.378764887855950169310935047772, −6.86128270965077973696608550181, −6.43369494193725334487730861939, −5.33578369533347663109905291754, −4.23681725318260795718280439817, −2.79082975054415023070699072984, −2.25417232854210174975124078679, −0.864611366487280071582463196077, 0.864611366487280071582463196077, 2.25417232854210174975124078679, 2.79082975054415023070699072984, 4.23681725318260795718280439817, 5.33578369533347663109905291754, 6.43369494193725334487730861939, 6.86128270965077973696608550181, 8.378764887855950169310935047772, 9.021408372978754718610141754146, 9.805353329081282244670300954756

Graph of the ZZ-function along the critical line