Properties

Degree 2
Conductor $ 2^{4} \cdot 251 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.201·3-s − 1.79·5-s − 3.34·7-s − 2.95·9-s + 5.67·11-s − 3.71·13-s − 0.363·15-s + 3.29·17-s + 0.297·19-s − 0.674·21-s − 8.29·23-s − 1.76·25-s − 1.20·27-s + 7.61·29-s − 7.30·31-s + 1.14·33-s + 6.00·35-s − 5.21·37-s − 0.750·39-s + 7.48·41-s + 0.751·43-s + 5.32·45-s − 7.73·47-s + 4.16·49-s + 0.665·51-s + 8.79·53-s − 10.2·55-s + ⋯
L(s)  = 1  + 0.116·3-s − 0.804·5-s − 1.26·7-s − 0.986·9-s + 1.71·11-s − 1.03·13-s − 0.0937·15-s + 0.799·17-s + 0.0681·19-s − 0.147·21-s − 1.72·23-s − 0.353·25-s − 0.231·27-s + 1.41·29-s − 1.31·31-s + 0.199·33-s + 1.01·35-s − 0.856·37-s − 0.120·39-s + 1.16·41-s + 0.114·43-s + 0.793·45-s − 1.12·47-s + 0.595·49-s + 0.0932·51-s + 1.20·53-s − 1.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4016\)    =    \(2^{4} \cdot 251\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9042203212$
$L(\frac12)$  $\approx$  $0.9042203212$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 - T \)
good3 \( 1 - 0.201T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 + 3.34T + 7T^{2} \)
11 \( 1 - 5.67T + 11T^{2} \)
13 \( 1 + 3.71T + 13T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 - 0.297T + 19T^{2} \)
23 \( 1 + 8.29T + 23T^{2} \)
29 \( 1 - 7.61T + 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 + 5.21T + 37T^{2} \)
41 \( 1 - 7.48T + 41T^{2} \)
43 \( 1 - 0.751T + 43T^{2} \)
47 \( 1 + 7.73T + 47T^{2} \)
53 \( 1 - 8.79T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 3.46T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 - 1.05T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 0.746T + 79T^{2} \)
83 \( 1 - 2.81T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 4.20T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.524863570603255433984935430565, −7.65499361041330866954034549933, −7.02779381803933182236549512535, −6.20763964419591628938368351282, −5.69813859736022923274866790193, −4.46713477089454983032317762169, −3.67866244673876919510519651488, −3.21280329582898592990572920398, −2.07380621700896475358818009666, −0.51711490602191842557264367422, 0.51711490602191842557264367422, 2.07380621700896475358818009666, 3.21280329582898592990572920398, 3.67866244673876919510519651488, 4.46713477089454983032317762169, 5.69813859736022923274866790193, 6.20763964419591628938368351282, 7.02779381803933182236549512535, 7.65499361041330866954034549933, 8.524863570603255433984935430565

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