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  + 2.29·3-s − 1.78·5-s − 0.671·7-s + 2.25·9-s − 2.23·11-s + 0.0525·13-s − 4.08·15-s + 5.52·17-s − 1.49·19-s − 1.53·21-s + 7.13·23-s − 1.81·25-s − 1.71·27-s + 2.08·29-s + 9.61·31-s − 5.11·33-s + 1.19·35-s + 5.33·37-s + 0.120·39-s − 11.8·41-s + 9.09·43-s − 4.01·45-s − 5.21·47-s − 6.54·49-s + 12.6·51-s + 4.27·53-s + 3.98·55-s + ⋯
L(s)  = 1  + 1.32·3-s − 0.798·5-s − 0.253·7-s + 0.750·9-s − 0.672·11-s + 0.0145·13-s − 1.05·15-s + 1.34·17-s − 0.343·19-s − 0.335·21-s + 1.48·23-s − 0.362·25-s − 0.330·27-s + 0.387·29-s + 1.72·31-s − 0.889·33-s + 0.202·35-s + 0.876·37-s + 0.0192·39-s − 1.85·41-s + 1.38·43-s − 0.598·45-s − 0.760·47-s − 0.935·49-s + 1.77·51-s + 0.587·53-s + 0.536·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.514088772$
$L(\frac12)$  $\approx$  $2.514088772$
$L(\frac{3}{2})$   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 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 + 1.78T + 5T^{2} \)
7 \( 1 + 0.671T + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 - 0.0525T + 13T^{2} \)
17 \( 1 - 5.52T + 17T^{2} \)
19 \( 1 + 1.49T + 19T^{2} \)
23 \( 1 - 7.13T + 23T^{2} \)
29 \( 1 - 2.08T + 29T^{2} \)
31 \( 1 - 9.61T + 31T^{2} \)
37 \( 1 - 5.33T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 - 4.27T + 53T^{2} \)
59 \( 1 - 0.163T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 8.24T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 7.86T + 79T^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−8.254071653684729429041688566602, −7.961509090725172282974859951960, −7.25733526524215019808054942274, −6.41020529170146097719042143799, −5.33300200037121164903276640614, −4.52080867505471646387412004877, −3.53342751799167033511759307183, −3.09607967857738804527961741729, −2.26210333199226590064341514490, −0.852472314786856807289469878206, 0.852472314786856807289469878206, 2.26210333199226590064341514490, 3.09607967857738804527961741729, 3.53342751799167033511759307183, 4.52080867505471646387412004877, 5.33300200037121164903276640614, 6.41020529170146097719042143799, 7.25733526524215019808054942274, 7.961509090725172282974859951960, 8.254071653684729429041688566602

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