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  − 0.452·3-s + 1.91·5-s − 1.96·7-s − 2.79·9-s − 0.787·11-s + 5.79·13-s − 0.864·15-s − 3.13·17-s − 4.63·19-s + 0.887·21-s − 2.51·23-s − 1.35·25-s + 2.62·27-s + 3.62·29-s + 6.20·31-s + 0.356·33-s − 3.74·35-s + 9.71·37-s − 2.62·39-s − 0.892·41-s + 5.80·43-s − 5.34·45-s − 3.65·47-s − 3.15·49-s + 1.41·51-s + 12.6·53-s − 1.50·55-s + ⋯
L(s)  = 1  − 0.261·3-s + 0.854·5-s − 0.741·7-s − 0.931·9-s − 0.237·11-s + 1.60·13-s − 0.223·15-s − 0.759·17-s − 1.06·19-s + 0.193·21-s − 0.523·23-s − 0.270·25-s + 0.504·27-s + 0.673·29-s + 1.11·31-s + 0.0620·33-s − 0.633·35-s + 1.59·37-s − 0.419·39-s − 0.139·41-s + 0.884·43-s − 0.796·45-s − 0.533·47-s − 0.450·49-s + 0.198·51-s + 1.73·53-s − 0.202·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.578786669$
$L(\frac12)$  $\approx$  $1.578786669$
$L(\frac{3}{2})$   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 \( 1 + 0.452T + 3T^{2} \)
5 \( 1 - 1.91T + 5T^{2} \)
7 \( 1 + 1.96T + 7T^{2} \)
11 \( 1 + 0.787T + 11T^{2} \)
13 \( 1 - 5.79T + 13T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 + 4.63T + 19T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 - 3.62T + 29T^{2} \)
31 \( 1 - 6.20T + 31T^{2} \)
37 \( 1 - 9.71T + 37T^{2} \)
41 \( 1 + 0.892T + 41T^{2} \)
43 \( 1 - 5.80T + 43T^{2} \)
47 \( 1 + 3.65T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + 6.08T + 59T^{2} \)
61 \( 1 - 6.47T + 61T^{2} \)
67 \( 1 - 8.94T + 67T^{2} \)
71 \( 1 + 7.14T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 4.55T + 83T^{2} \)
89 \( 1 - 1.55T + 89T^{2} \)
97 \( 1 - 5.55T + 97T^{2} \)
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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

−8.645404602150212356058185276131, −7.84566398980557060482125804147, −6.57277997339417304409813505374, −6.15906886268005419239879683793, −5.84727189562920906707799813201, −4.69911389018877723208091209085, −3.83513378810614359075778006193, −2.84107858550559946226593890461, −2.08790382222521912204489111129, −0.71233223546209213725545763149, 0.71233223546209213725545763149, 2.08790382222521912204489111129, 2.84107858550559946226593890461, 3.83513378810614359075778006193, 4.69911389018877723208091209085, 5.84727189562920906707799813201, 6.15906886268005419239879683793, 6.57277997339417304409813505374, 7.84566398980557060482125804147, 8.645404602150212356058185276131

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