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  + 2.21·3-s + 3.09·5-s + 4.67·7-s + 1.90·9-s + 4.76·11-s − 4.18·13-s + 6.86·15-s − 2.76·17-s + 4.43·19-s + 10.3·21-s − 8.88·23-s + 4.59·25-s − 2.41·27-s + 2.14·29-s − 7.46·31-s + 10.5·33-s + 14.4·35-s + 9.30·37-s − 9.26·39-s − 8.39·41-s + 3.77·43-s + 5.90·45-s + 3.41·47-s + 14.9·49-s − 6.12·51-s + 11.5·53-s + 14.7·55-s + ⋯
L(s)  = 1  + 1.27·3-s + 1.38·5-s + 1.76·7-s + 0.635·9-s + 1.43·11-s − 1.15·13-s + 1.77·15-s − 0.670·17-s + 1.01·19-s + 2.26·21-s − 1.85·23-s + 0.918·25-s − 0.465·27-s + 0.399·29-s − 1.34·31-s + 1.83·33-s + 2.44·35-s + 1.52·37-s − 1.48·39-s − 1.31·41-s + 0.576·43-s + 0.880·45-s + 0.498·47-s + 2.12·49-s − 0.857·51-s + 1.58·53-s + 1.98·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.944312759$
$L(\frac12)$  $\approx$  $4.944312759$
$L(\frac{3}{2})$   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 \( 1 - 2.21T + 3T^{2} \)
5 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 - 4.67T + 7T^{2} \)
11 \( 1 - 4.76T + 11T^{2} \)
13 \( 1 + 4.18T + 13T^{2} \)
17 \( 1 + 2.76T + 17T^{2} \)
19 \( 1 - 4.43T + 19T^{2} \)
23 \( 1 + 8.88T + 23T^{2} \)
29 \( 1 - 2.14T + 29T^{2} \)
31 \( 1 + 7.46T + 31T^{2} \)
37 \( 1 - 9.30T + 37T^{2} \)
41 \( 1 + 8.39T + 41T^{2} \)
43 \( 1 - 3.77T + 43T^{2} \)
47 \( 1 - 3.41T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 0.620T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 3.88T + 67T^{2} \)
71 \( 1 + 5.36T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 + 4.20T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 9.25T + 97T^{2} \)
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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

−8.692194052824346285441798736820, −7.68606378248441816697934665597, −7.31258024649243782930419194027, −6.14459883107520123108288231423, −5.48179443193174163546726806250, −4.56860823415893187339225861052, −3.89915870015030187047997275902, −2.61924803759791608951568042854, −1.98291349279928036952712806466, −1.45476707413079181268051807104, 1.45476707413079181268051807104, 1.98291349279928036952712806466, 2.61924803759791608951568042854, 3.89915870015030187047997275902, 4.56860823415893187339225861052, 5.48179443193174163546726806250, 6.14459883107520123108288231423, 7.31258024649243782930419194027, 7.68606378248441816697934665597, 8.692194052824346285441798736820

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