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  + 0.0812·3-s + 4.07·5-s − 4.55·7-s − 2.99·9-s − 3.53·11-s − 1.88·13-s + 0.331·15-s + 7.25·17-s + 6.82·19-s − 0.370·21-s + 0.510·23-s + 11.6·25-s − 0.487·27-s − 0.891·29-s − 4.69·31-s − 0.287·33-s − 18.5·35-s + 5.85·37-s − 0.153·39-s − 10.5·41-s + 5.97·43-s − 12.2·45-s + 3.59·47-s + 13.7·49-s + 0.589·51-s − 7.13·53-s − 14.4·55-s + ⋯
L(s)  = 1  + 0.0469·3-s + 1.82·5-s − 1.72·7-s − 0.997·9-s − 1.06·11-s − 0.523·13-s + 0.0856·15-s + 1.75·17-s + 1.56·19-s − 0.0808·21-s + 0.106·23-s + 2.32·25-s − 0.0937·27-s − 0.165·29-s − 0.843·31-s − 0.0499·33-s − 3.14·35-s + 0.962·37-s − 0.0245·39-s − 1.64·41-s + 0.911·43-s − 1.82·45-s + 0.524·47-s + 1.96·49-s + 0.0825·51-s − 0.980·53-s − 1.94·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.930932988$
$L(\frac12)$  $\approx$  $1.930932988$
$L(\frac{3}{2})$   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 \( 1 - 0.0812T + 3T^{2} \)
5 \( 1 - 4.07T + 5T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
11 \( 1 + 3.53T + 11T^{2} \)
13 \( 1 + 1.88T + 13T^{2} \)
17 \( 1 - 7.25T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 - 0.510T + 23T^{2} \)
29 \( 1 + 0.891T + 29T^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 - 5.85T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 5.97T + 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + 7.13T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 5.92T + 61T^{2} \)
67 \( 1 - 8.31T + 67T^{2} \)
71 \( 1 - 8.26T + 71T^{2} \)
73 \( 1 + 9.74T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 8.97T + 83T^{2} \)
89 \( 1 - 3.15T + 89T^{2} \)
97 \( 1 + 5.31T + 97T^{2} \)
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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

−8.615085708514932251167838803540, −7.60065974045566611652246118128, −6.90287390599893286375113633804, −5.98365242557309944881614598820, −5.56212661551102388771306612903, −5.18866603691428358162128419660, −3.35646277963571232808852849666, −2.98588632531577016856792975716, −2.20395001441055430474128900652, −0.77049089389587631588947243599, 0.77049089389587631588947243599, 2.20395001441055430474128900652, 2.98588632531577016856792975716, 3.35646277963571232808852849666, 5.18866603691428358162128419660, 5.56212661551102388771306612903, 5.98365242557309944881614598820, 6.90287390599893286375113633804, 7.60065974045566611652246118128, 8.615085708514932251167838803540

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