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  + 1.28·3-s + 3.72·5-s + 0.978·7-s − 1.35·9-s − 0.158·11-s + 0.124·13-s + 4.77·15-s + 0.487·17-s − 4.09·19-s + 1.25·21-s + 5.58·23-s + 8.84·25-s − 5.58·27-s + 9.50·29-s + 4.53·31-s − 0.203·33-s + 3.64·35-s − 4.70·37-s + 0.159·39-s + 8.31·41-s − 12.3·43-s − 5.02·45-s − 4.37·47-s − 6.04·49-s + 0.625·51-s + 10.3·53-s − 0.591·55-s + ⋯
L(s)  = 1  + 0.741·3-s + 1.66·5-s + 0.369·7-s − 0.450·9-s − 0.0478·11-s + 0.0344·13-s + 1.23·15-s + 0.118·17-s − 0.939·19-s + 0.274·21-s + 1.16·23-s + 1.76·25-s − 1.07·27-s + 1.76·29-s + 0.814·31-s − 0.0355·33-s + 0.615·35-s − 0.773·37-s + 0.0255·39-s + 1.29·41-s − 1.88·43-s − 0.749·45-s − 0.637·47-s − 0.863·49-s + 0.0875·51-s + 1.42·53-s − 0.0797·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.596981614$
$L(\frac12)$  $\approx$  $3.596981614$
$L(\frac{3}{2})$   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 \( 1 - 1.28T + 3T^{2} \)
5 \( 1 - 3.72T + 5T^{2} \)
7 \( 1 - 0.978T + 7T^{2} \)
11 \( 1 + 0.158T + 11T^{2} \)
13 \( 1 - 0.124T + 13T^{2} \)
17 \( 1 - 0.487T + 17T^{2} \)
19 \( 1 + 4.09T + 19T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 - 9.50T + 29T^{2} \)
31 \( 1 - 4.53T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 - 8.31T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + 4.37T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 6.42T + 59T^{2} \)
61 \( 1 - 6.11T + 61T^{2} \)
67 \( 1 - 1.25T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + 5.47T + 89T^{2} \)
97 \( 1 - 5.60T + 97T^{2} \)
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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

−8.473231078090928279568005923966, −8.035019520204371562805906369748, −6.62412628688320675804588306694, −6.49763227850534186648002457950, −5.30874626283774510624501060277, −4.96417575145308193915245599398, −3.66764148383235302294331620088, −2.64939300673144992796115874937, −2.21606508114554400620053332015, −1.11378111298221052593387432073, 1.11378111298221052593387432073, 2.21606508114554400620053332015, 2.64939300673144992796115874937, 3.66764148383235302294331620088, 4.96417575145308193915245599398, 5.30874626283774510624501060277, 6.49763227850534186648002457950, 6.62412628688320675804588306694, 8.035019520204371562805906369748, 8.473231078090928279568005923966

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