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  − 0.259·3-s − 1.96·5-s + 4.76·7-s − 2.93·9-s − 3.31·11-s + 0.493·13-s + 0.509·15-s − 3.02·17-s − 1.99·19-s − 1.23·21-s − 3.65·23-s − 1.14·25-s + 1.53·27-s + 5.29·29-s + 8.88·31-s + 0.859·33-s − 9.35·35-s + 6.88·37-s − 0.127·39-s − 3.94·41-s − 10.3·43-s + 5.75·45-s + 12.7·47-s + 15.6·49-s + 0.785·51-s + 5.75·53-s + 6.50·55-s + ⋯
L(s)  = 1  − 0.149·3-s − 0.878·5-s + 1.80·7-s − 0.977·9-s − 0.998·11-s + 0.136·13-s + 0.131·15-s − 0.734·17-s − 0.457·19-s − 0.269·21-s − 0.761·23-s − 0.228·25-s + 0.296·27-s + 0.983·29-s + 1.59·31-s + 0.149·33-s − 1.58·35-s + 1.13·37-s − 0.0204·39-s − 0.615·41-s − 1.58·43-s + 0.858·45-s + 1.86·47-s + 2.24·49-s + 0.109·51-s + 0.790·53-s + 0.877·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.353462505$
$L(\frac12)$  $\approx$  $1.353462505$
$L(\frac{3}{2})$   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 \( 1 + 0.259T + 3T^{2} \)
5 \( 1 + 1.96T + 5T^{2} \)
7 \( 1 - 4.76T + 7T^{2} \)
11 \( 1 + 3.31T + 11T^{2} \)
13 \( 1 - 0.493T + 13T^{2} \)
17 \( 1 + 3.02T + 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 - 8.88T + 31T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 + 3.94T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 5.75T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 8.88T + 73T^{2} \)
79 \( 1 + 8.79T + 79T^{2} \)
83 \( 1 + 8.70T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−8.420763532149976900588170946107, −7.937809165355605386334783102158, −7.17630162114432478198656773295, −6.12954716429041824351954163706, −5.36459406205643971763069471367, −4.62298581930946546717264084744, −4.11284793793116013737869822164, −2.81364816474457901127193573354, −2.07062856396230468854062154361, −0.65705871577796517167448261819, 0.65705871577796517167448261819, 2.07062856396230468854062154361, 2.81364816474457901127193573354, 4.11284793793116013737869822164, 4.62298581930946546717264084744, 5.36459406205643971763069471367, 6.12954716429041824351954163706, 7.17630162114432478198656773295, 7.937809165355605386334783102158, 8.420763532149976900588170946107

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