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  − 1.96·3-s + 0.807·5-s − 3.34·7-s + 0.848·9-s − 4.04·11-s − 4.44·13-s − 1.58·15-s − 5.91·17-s − 2.98·19-s + 6.55·21-s − 7.45·23-s − 4.34·25-s + 4.22·27-s + 7.98·29-s − 9.58·31-s + 7.93·33-s − 2.69·35-s + 2.67·37-s + 8.71·39-s + 3.85·41-s − 9.57·43-s + 0.685·45-s + 9.46·47-s + 4.17·49-s + 11.6·51-s − 9.89·53-s − 3.26·55-s + ⋯
L(s)  = 1  − 1.13·3-s + 0.361·5-s − 1.26·7-s + 0.282·9-s − 1.21·11-s − 1.23·13-s − 0.408·15-s − 1.43·17-s − 0.684·19-s + 1.43·21-s − 1.55·23-s − 0.869·25-s + 0.812·27-s + 1.48·29-s − 1.72·31-s + 1.38·33-s − 0.456·35-s + 0.440·37-s + 1.39·39-s + 0.601·41-s − 1.46·43-s + 0.102·45-s + 1.38·47-s + 0.596·49-s + 1.62·51-s − 1.35·53-s − 0.440·55-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  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.01286480563$
$L(\frac12)$  $\approx$  $0.01286480563$
$L(\frac{3}{2})$   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 \( 1 + 1.96T + 3T^{2} \)
5 \( 1 - 0.807T + 5T^{2} \)
7 \( 1 + 3.34T + 7T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + 5.91T + 17T^{2} \)
19 \( 1 + 2.98T + 19T^{2} \)
23 \( 1 + 7.45T + 23T^{2} \)
29 \( 1 - 7.98T + 29T^{2} \)
31 \( 1 + 9.58T + 31T^{2} \)
37 \( 1 - 2.67T + 37T^{2} \)
41 \( 1 - 3.85T + 41T^{2} \)
43 \( 1 + 9.57T + 43T^{2} \)
47 \( 1 - 9.46T + 47T^{2} \)
53 \( 1 + 9.89T + 53T^{2} \)
59 \( 1 + 6.88T + 59T^{2} \)
61 \( 1 + 7.56T + 61T^{2} \)
67 \( 1 - 8.97T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 3.51T + 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 + 9.42T + 83T^{2} \)
89 \( 1 - 0.414T + 89T^{2} \)
97 \( 1 - 8.39T + 97T^{2} \)
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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

−8.387553926275176660299772274149, −7.61786558539588270078420376976, −6.62710159528752554205534048099, −6.32457208511920033614983438780, −5.52995698523825272866225685408, −4.87932414431251299362258435499, −4.00810009658666076631985605504, −2.75743034608549314679703088660, −2.12271695362634481158367496367, −0.06240768949309930971462466713, 0.06240768949309930971462466713, 2.12271695362634481158367496367, 2.75743034608549314679703088660, 4.00810009658666076631985605504, 4.87932414431251299362258435499, 5.52995698523825272866225685408, 6.32457208511920033614983438780, 6.62710159528752554205534048099, 7.61786558539588270078420376976, 8.387553926275176660299772274149

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