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  − 2.18·3-s − 1.66·5-s + 1.81·7-s + 1.76·9-s + 3.97·11-s − 4.88·13-s + 3.63·15-s − 3.09·17-s + 3.71·19-s − 3.96·21-s − 2.52·23-s − 2.22·25-s + 2.70·27-s + 4.16·29-s − 4.17·31-s − 8.67·33-s − 3.02·35-s + 11.0·37-s + 10.6·39-s − 0.0987·41-s + 2.04·43-s − 2.93·45-s − 0.469·47-s − 3.69·49-s + 6.75·51-s − 3.85·53-s − 6.62·55-s + ⋯
L(s)  = 1  − 1.25·3-s − 0.745·5-s + 0.686·7-s + 0.587·9-s + 1.19·11-s − 1.35·13-s + 0.939·15-s − 0.750·17-s + 0.853·19-s − 0.865·21-s − 0.525·23-s − 0.444·25-s + 0.519·27-s + 0.772·29-s − 0.750·31-s − 1.51·33-s − 0.512·35-s + 1.81·37-s + 1.70·39-s − 0.0154·41-s + 0.312·43-s − 0.437·45-s − 0.0684·47-s − 0.528·49-s + 0.946·51-s − 0.530·53-s − 0.893·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8133126253$
$L(\frac12)$  $\approx$  $0.8133126253$
$L(\frac{3}{2})$   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 \( 1 + 2.18T + 3T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 - 3.97T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 3.09T + 17T^{2} \)
19 \( 1 - 3.71T + 19T^{2} \)
23 \( 1 + 2.52T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 0.0987T + 41T^{2} \)
43 \( 1 - 2.04T + 43T^{2} \)
47 \( 1 + 0.469T + 47T^{2} \)
53 \( 1 + 3.85T + 53T^{2} \)
59 \( 1 + 7.09T + 59T^{2} \)
61 \( 1 + 0.640T + 61T^{2} \)
67 \( 1 + 1.93T + 67T^{2} \)
71 \( 1 + 0.867T + 71T^{2} \)
73 \( 1 + 6.51T + 73T^{2} \)
79 \( 1 - 9.82T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 9.81T + 89T^{2} \)
97 \( 1 + 1.98T + 97T^{2} \)
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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

−8.290859415889488888414749026149, −7.60205576513450234333308839614, −6.95754898396719854932526685134, −6.20068000089059005164426222289, −5.47760951103174953285297786067, −4.55110644535511700384123297887, −4.29614596372741741860012632818, −3.00179615210389152173555874428, −1.73244615216372368272700576123, −0.55647462105700252076438897002, 0.55647462105700252076438897002, 1.73244615216372368272700576123, 3.00179615210389152173555874428, 4.29614596372741741860012632818, 4.55110644535511700384123297887, 5.47760951103174953285297786067, 6.20068000089059005164426222289, 6.95754898396719854932526685134, 7.60205576513450234333308839614, 8.290859415889488888414749026149

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