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  − 1.63·3-s + 2.87·5-s + 2.53·7-s − 0.317·9-s + 3.58·11-s + 5.49·13-s − 4.70·15-s + 6.64·17-s + 3.16·19-s − 4.15·21-s − 2.23·23-s + 3.25·25-s + 5.43·27-s + 5.29·29-s − 6.91·31-s − 5.86·33-s + 7.28·35-s + 6.97·37-s − 9.00·39-s + 8.56·41-s − 7.68·43-s − 0.912·45-s + 2.74·47-s − 0.577·49-s − 10.8·51-s − 6.46·53-s + 10.2·55-s + ⋯
L(s)  = 1  − 0.945·3-s + 1.28·5-s + 0.957·7-s − 0.105·9-s + 1.08·11-s + 1.52·13-s − 1.21·15-s + 1.61·17-s + 0.726·19-s − 0.905·21-s − 0.465·23-s + 0.651·25-s + 1.04·27-s + 0.983·29-s − 1.24·31-s − 1.02·33-s + 1.23·35-s + 1.14·37-s − 1.44·39-s + 1.33·41-s − 1.17·43-s − 0.136·45-s + 0.400·47-s − 0.0825·49-s − 1.52·51-s − 0.887·53-s + 1.38·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.518236444$
$L(\frac12)$  $\approx$  $2.518236444$
$L(\frac{3}{2})$   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 \( 1 + 1.63T + 3T^{2} \)
5 \( 1 - 2.87T + 5T^{2} \)
7 \( 1 - 2.53T + 7T^{2} \)
11 \( 1 - 3.58T + 11T^{2} \)
13 \( 1 - 5.49T + 13T^{2} \)
17 \( 1 - 6.64T + 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 6.91T + 31T^{2} \)
37 \( 1 - 6.97T + 37T^{2} \)
41 \( 1 - 8.56T + 41T^{2} \)
43 \( 1 + 7.68T + 43T^{2} \)
47 \( 1 - 2.74T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + 2.83T + 59T^{2} \)
61 \( 1 + 1.72T + 61T^{2} \)
67 \( 1 + 6.56T + 67T^{2} \)
71 \( 1 + 8.23T + 71T^{2} \)
73 \( 1 - 3.78T + 73T^{2} \)
79 \( 1 + 1.99T + 79T^{2} \)
83 \( 1 - 3.83T + 83T^{2} \)
89 \( 1 + 2.50T + 89T^{2} \)
97 \( 1 + 19.1T + 97T^{2} \)
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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

−8.447812963050814930478629248293, −7.76791777090177741639848718985, −6.68629665119724066035406572397, −5.93229204559247803840337919227, −5.75427025141431172189907468826, −4.94065453329176072183166579304, −3.94731934012417733694528315789, −2.93082965242902212954169420030, −1.50182027629641387545992719374, −1.16666374012568492408831229437, 1.16666374012568492408831229437, 1.50182027629641387545992719374, 2.93082965242902212954169420030, 3.94731934012417733694528315789, 4.94065453329176072183166579304, 5.75427025141431172189907468826, 5.93229204559247803840337919227, 6.68629665119724066035406572397, 7.76791777090177741639848718985, 8.447812963050814930478629248293

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