Properties

Label 2-1008-1.1-c5-0-34
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 77.6·5-s + 49·7-s + 477.·11-s − 63.7·13-s − 1.03e3·17-s + 667.·19-s + 3.25e3·23-s + 2.90e3·25-s − 2.30e3·29-s − 3.71e3·31-s + 3.80e3·35-s + 1.22e4·37-s + 1.82e3·41-s + 2.07e4·43-s − 4.28e3·47-s + 2.40e3·49-s − 2.57e4·53-s + 3.70e4·55-s − 2.83e3·59-s + 1.68e4·61-s − 4.94e3·65-s + 6.25e4·67-s + 7.23e4·71-s − 5.56e4·73-s + 2.33e4·77-s + 3.98e3·79-s − 4.60e4·83-s + ⋯
L(s)  = 1  + 1.38·5-s + 0.377·7-s + 1.18·11-s − 0.104·13-s − 0.870·17-s + 0.423·19-s + 1.28·23-s + 0.928·25-s − 0.508·29-s − 0.694·31-s + 0.524·35-s + 1.47·37-s + 0.169·41-s + 1.71·43-s − 0.282·47-s + 0.142·49-s − 1.25·53-s + 1.65·55-s − 0.106·59-s + 0.578·61-s − 0.145·65-s + 1.70·67-s + 1.70·71-s − 1.22·73-s + 0.449·77-s + 0.0719·79-s − 0.734·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.840998789\)
\(L(\frac12)\) \(\approx\) \(3.840998789\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 49T \)
good5 \( 1 - 77.6T + 3.12e3T^{2} \)
11 \( 1 - 477.T + 1.61e5T^{2} \)
13 \( 1 + 63.7T + 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 - 667.T + 2.47e6T^{2} \)
23 \( 1 - 3.25e3T + 6.43e6T^{2} \)
29 \( 1 + 2.30e3T + 2.05e7T^{2} \)
31 \( 1 + 3.71e3T + 2.86e7T^{2} \)
37 \( 1 - 1.22e4T + 6.93e7T^{2} \)
41 \( 1 - 1.82e3T + 1.15e8T^{2} \)
43 \( 1 - 2.07e4T + 1.47e8T^{2} \)
47 \( 1 + 4.28e3T + 2.29e8T^{2} \)
53 \( 1 + 2.57e4T + 4.18e8T^{2} \)
59 \( 1 + 2.83e3T + 7.14e8T^{2} \)
61 \( 1 - 1.68e4T + 8.44e8T^{2} \)
67 \( 1 - 6.25e4T + 1.35e9T^{2} \)
71 \( 1 - 7.23e4T + 1.80e9T^{2} \)
73 \( 1 + 5.56e4T + 2.07e9T^{2} \)
79 \( 1 - 3.98e3T + 3.07e9T^{2} \)
83 \( 1 + 4.60e4T + 3.93e9T^{2} \)
89 \( 1 + 1.35e5T + 5.58e9T^{2} \)
97 \( 1 - 1.42e5T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.365106844464828524164077208610, −8.660242259122220289873101619703, −7.42586523851789983103533892509, −6.59041533299849448563661684845, −5.86717995133336860686690317105, −4.99667737141553815695978103341, −4.00134588714583596209304140489, −2.69116999021782845940189815854, −1.77802965902542530348386408135, −0.905341915206997831297072091040, 0.905341915206997831297072091040, 1.77802965902542530348386408135, 2.69116999021782845940189815854, 4.00134588714583596209304140489, 4.99667737141553815695978103341, 5.86717995133336860686690317105, 6.59041533299849448563661684845, 7.42586523851789983103533892509, 8.660242259122220289873101619703, 9.365106844464828524164077208610

Graph of the $Z$-function along the critical line