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  + 2.13·3-s + 0.257·5-s − 0.578·7-s + 1.54·9-s + 1.44·11-s + 1.23·13-s + 0.548·15-s + 3.22·17-s + 2.16·19-s − 1.23·21-s + 3.59·23-s − 4.93·25-s − 3.10·27-s + 1.34·29-s − 2.91·31-s + 3.07·33-s − 0.148·35-s + 7.43·37-s + 2.62·39-s + 9.61·41-s + 4.00·43-s + 0.397·45-s + 7.55·47-s − 6.66·49-s + 6.88·51-s − 5.08·53-s + 0.371·55-s + ⋯
L(s)  = 1  + 1.23·3-s + 0.115·5-s − 0.218·7-s + 0.514·9-s + 0.435·11-s + 0.342·13-s + 0.141·15-s + 0.783·17-s + 0.496·19-s − 0.268·21-s + 0.748·23-s − 0.986·25-s − 0.597·27-s + 0.248·29-s − 0.523·31-s + 0.535·33-s − 0.0251·35-s + 1.22·37-s + 0.420·39-s + 1.50·41-s + 0.610·43-s + 0.0592·45-s + 1.10·47-s − 0.952·49-s + 0.964·51-s − 0.698·53-s + 0.0500·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.247840773$
$L(\frac12)$  $\approx$  $3.247840773$
$L(\frac{3}{2})$   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 \( 1 - 2.13T + 3T^{2} \)
5 \( 1 - 0.257T + 5T^{2} \)
7 \( 1 + 0.578T + 7T^{2} \)
11 \( 1 - 1.44T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 3.22T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 - 3.59T + 23T^{2} \)
29 \( 1 - 1.34T + 29T^{2} \)
31 \( 1 + 2.91T + 31T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 - 9.61T + 41T^{2} \)
43 \( 1 - 4.00T + 43T^{2} \)
47 \( 1 - 7.55T + 47T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 - 4.79T + 59T^{2} \)
61 \( 1 + 0.119T + 61T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + 9.70T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−8.466962391563079260070640493661, −7.73063472883338515216100332070, −7.29235937157878090851391258419, −6.17868459331718566039952510336, −5.60691913304579515122358351869, −4.43852535700573488647234130354, −3.65172019864347158540738578495, −3.00163687324987865973890406689, −2.15047859355783359128818561100, −1.01873155319082119995949406872, 1.01873155319082119995949406872, 2.15047859355783359128818561100, 3.00163687324987865973890406689, 3.65172019864347158540738578495, 4.43852535700573488647234130354, 5.60691913304579515122358351869, 6.17868459331718566039952510336, 7.29235937157878090851391258419, 7.73063472883338515216100332070, 8.466962391563079260070640493661

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