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  − 3.32·3-s − 0.316·5-s − 0.268·7-s + 8.06·9-s + 0.366·11-s − 3.69·13-s + 1.05·15-s + 5.91·17-s − 7.61·19-s + 0.891·21-s + 6.61·23-s − 4.89·25-s − 16.8·27-s + 5.48·29-s + 4.15·31-s − 1.21·33-s + 0.0849·35-s + 8.30·37-s + 12.2·39-s + 3.92·41-s − 6.01·43-s − 2.55·45-s − 1.79·47-s − 6.92·49-s − 19.6·51-s − 10.5·53-s − 0.116·55-s + ⋯
L(s)  = 1  − 1.92·3-s − 0.141·5-s − 0.101·7-s + 2.68·9-s + 0.110·11-s − 1.02·13-s + 0.272·15-s + 1.43·17-s − 1.74·19-s + 0.194·21-s + 1.37·23-s − 0.979·25-s − 3.23·27-s + 1.01·29-s + 0.746·31-s − 0.212·33-s + 0.0143·35-s + 1.36·37-s + 1.96·39-s + 0.612·41-s − 0.917·43-s − 0.380·45-s − 0.261·47-s − 0.989·49-s − 2.75·51-s − 1.44·53-s − 0.0156·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6850768891$
$L(\frac12)$  $\approx$  $0.6850768891$
$L(\frac{3}{2})$   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 \( 1 + 3.32T + 3T^{2} \)
5 \( 1 + 0.316T + 5T^{2} \)
7 \( 1 + 0.268T + 7T^{2} \)
11 \( 1 - 0.366T + 11T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 + 7.61T + 19T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 - 5.48T + 29T^{2} \)
31 \( 1 - 4.15T + 31T^{2} \)
37 \( 1 - 8.30T + 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 + 6.01T + 43T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 - 3.99T + 59T^{2} \)
61 \( 1 - 1.88T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + 8.15T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + 7.93T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 1.39T + 89T^{2} \)
97 \( 1 + 9.53T + 97T^{2} \)
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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

−8.251796062512354376560746961982, −7.51453526525806909739370748338, −6.76696967766435751551203258765, −6.20513495284926245126379694823, −5.54226727408373191807640507526, −4.68999795946863503743706550453, −4.32771958548964607759994969339, −2.98526362922887982810856564878, −1.60512932312360658064922768746, −0.54058863156484521598132696001, 0.54058863156484521598132696001, 1.60512932312360658064922768746, 2.98526362922887982810856564878, 4.32771958548964607759994969339, 4.68999795946863503743706550453, 5.54226727408373191807640507526, 6.20513495284926245126379694823, 6.76696967766435751551203258765, 7.51453526525806909739370748338, 8.251796062512354376560746961982

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