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  + 1.19·3-s − 1.76·5-s + 4.20·7-s − 1.57·9-s + 6.20·11-s + 6.78·13-s − 2.11·15-s − 4.29·17-s + 2.98·19-s + 5.02·21-s + 4.21·23-s − 1.87·25-s − 5.46·27-s + 0.933·29-s + 4.70·31-s + 7.40·33-s − 7.43·35-s − 11.8·37-s + 8.09·39-s + 7.43·41-s + 7.15·43-s + 2.78·45-s − 11.7·47-s + 10.6·49-s − 5.13·51-s + 4.02·53-s − 10.9·55-s + ⋯
L(s)  = 1  + 0.689·3-s − 0.790·5-s + 1.58·7-s − 0.524·9-s + 1.86·11-s + 1.88·13-s − 0.544·15-s − 1.04·17-s + 0.684·19-s + 1.09·21-s + 0.878·23-s − 0.375·25-s − 1.05·27-s + 0.173·29-s + 0.844·31-s + 1.28·33-s − 1.25·35-s − 1.94·37-s + 1.29·39-s + 1.16·41-s + 1.09·43-s + 0.414·45-s − 1.70·47-s + 1.52·49-s − 0.718·51-s + 0.552·53-s − 1.47·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.064414224$
$L(\frac12)$  $\approx$  $3.064414224$
$L(\frac{3}{2})$   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 \( 1 - 1.19T + 3T^{2} \)
5 \( 1 + 1.76T + 5T^{2} \)
7 \( 1 - 4.20T + 7T^{2} \)
11 \( 1 - 6.20T + 11T^{2} \)
13 \( 1 - 6.78T + 13T^{2} \)
17 \( 1 + 4.29T + 17T^{2} \)
19 \( 1 - 2.98T + 19T^{2} \)
23 \( 1 - 4.21T + 23T^{2} \)
29 \( 1 - 0.933T + 29T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 - 7.43T + 41T^{2} \)
43 \( 1 - 7.15T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 4.02T + 53T^{2} \)
59 \( 1 + 9.10T + 59T^{2} \)
61 \( 1 - 3.77T + 61T^{2} \)
67 \( 1 - 9.44T + 67T^{2} \)
71 \( 1 - 3.31T + 71T^{2} \)
73 \( 1 + 6.72T + 73T^{2} \)
79 \( 1 + 1.40T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 6.83T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−8.593421209428342299588891267486, −7.962518560543930735241520221632, −7.08091358968983013628345876868, −6.33467472319977717444403843037, −5.43334218225435278074148421640, −4.36339207905413495486378246084, −3.91736989177573915002616914304, −3.12874089153698072058436243801, −1.80436330526890416089463946996, −1.09262056208496493735861799200, 1.09262056208496493735861799200, 1.80436330526890416089463946996, 3.12874089153698072058436243801, 3.91736989177573915002616914304, 4.36339207905413495486378246084, 5.43334218225435278074148421640, 6.33467472319977717444403843037, 7.08091358968983013628345876868, 7.962518560543930735241520221632, 8.593421209428342299588891267486

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