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  − 0.358·3-s + 2.25·5-s + 4.66·7-s − 2.87·9-s + 0.681·11-s + 0.464·13-s − 0.806·15-s − 0.986·17-s + 2.32·19-s − 1.67·21-s + 5.37·23-s + 0.0786·25-s + 2.10·27-s − 4.75·29-s + 1.57·31-s − 0.244·33-s + 10.5·35-s + 3.55·37-s − 0.166·39-s + 0.278·41-s + 1.78·43-s − 6.47·45-s − 2.22·47-s + 14.8·49-s + 0.353·51-s + 2.88·53-s + 1.53·55-s + ⋯
L(s)  = 1  − 0.206·3-s + 1.00·5-s + 1.76·7-s − 0.957·9-s + 0.205·11-s + 0.128·13-s − 0.208·15-s − 0.239·17-s + 0.534·19-s − 0.364·21-s + 1.12·23-s + 0.0157·25-s + 0.404·27-s − 0.882·29-s + 0.282·31-s − 0.0424·33-s + 1.77·35-s + 0.583·37-s − 0.0266·39-s + 0.0434·41-s + 0.271·43-s − 0.964·45-s − 0.324·47-s + 2.11·49-s + 0.0494·51-s + 0.396·53-s + 0.207·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.692094670$
$L(\frac12)$  $\approx$  $2.692094670$
$L(\frac{3}{2})$   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 \( 1 + 0.358T + 3T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 - 4.66T + 7T^{2} \)
11 \( 1 - 0.681T + 11T^{2} \)
13 \( 1 - 0.464T + 13T^{2} \)
17 \( 1 + 0.986T + 17T^{2} \)
19 \( 1 - 2.32T + 19T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 - 1.57T + 31T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 - 0.278T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + 2.22T + 47T^{2} \)
53 \( 1 - 2.88T + 53T^{2} \)
59 \( 1 - 6.32T + 59T^{2} \)
61 \( 1 - 0.0560T + 61T^{2} \)
67 \( 1 - 5.83T + 67T^{2} \)
71 \( 1 - 2.74T + 71T^{2} \)
73 \( 1 - 9.63T + 73T^{2} \)
79 \( 1 + 3.00T + 79T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 4.01T + 97T^{2} \)
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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

−8.462777490928147695927860672653, −7.82388421993055696461020369478, −6.97368543171166204561112604128, −6.05595719468598832959016683558, −5.37011798282942996151875459932, −4.99163882249833067234353573160, −3.94839022244706069996338857090, −2.71919105990402096771452854513, −1.93515393976980912760580319735, −1.01493568557011409454043413181, 1.01493568557011409454043413181, 1.93515393976980912760580319735, 2.71919105990402096771452854513, 3.94839022244706069996338857090, 4.99163882249833067234353573160, 5.37011798282942996151875459932, 6.05595719468598832959016683558, 6.97368543171166204561112604128, 7.82388421993055696461020369478, 8.462777490928147695927860672653

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