Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.34·3-s + 2.39·5-s + 3.54·7-s + 2.50·9-s + 4.15·11-s + 3.77·13-s + 5.62·15-s − 5.89·17-s + 2.10·19-s + 8.32·21-s + 0.956·23-s + 0.741·25-s − 1.16·27-s − 10.1·29-s + 1.81·31-s + 9.73·33-s + 8.50·35-s + 2.63·37-s + 8.85·39-s − 2.88·41-s + 9.00·43-s + 5.99·45-s − 2.83·47-s + 5.59·49-s − 13.8·51-s − 7.26·53-s + 9.94·55-s + ⋯
L(s)  = 1  + 1.35·3-s + 1.07·5-s + 1.34·7-s + 0.834·9-s + 1.25·11-s + 1.04·13-s + 1.45·15-s − 1.42·17-s + 0.483·19-s + 1.81·21-s + 0.199·23-s + 0.148·25-s − 0.224·27-s − 1.87·29-s + 0.326·31-s + 1.69·33-s + 1.43·35-s + 0.433·37-s + 1.41·39-s − 0.450·41-s + 1.37·43-s + 0.893·45-s − 0.413·47-s + 0.798·49-s − 1.93·51-s − 0.997·53-s + 1.34·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.368744774$
$L(\frac12)$  $\approx$  $5.368744774$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;751\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;751\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 2.34T + 3T^{2} \)
5 \( 1 - 2.39T + 5T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
23 \( 1 - 0.956T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 1.81T + 31T^{2} \)
37 \( 1 - 2.63T + 37T^{2} \)
41 \( 1 + 2.88T + 41T^{2} \)
43 \( 1 - 9.00T + 43T^{2} \)
47 \( 1 + 2.83T + 47T^{2} \)
53 \( 1 + 7.26T + 53T^{2} \)
59 \( 1 - 2.36T + 59T^{2} \)
61 \( 1 - 3.81T + 61T^{2} \)
67 \( 1 - 6.00T + 67T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 - 2.69T + 73T^{2} \)
79 \( 1 + 1.14T + 79T^{2} \)
83 \( 1 - 0.157T + 83T^{2} \)
89 \( 1 - 3.98T + 89T^{2} \)
97 \( 1 + 3.46T + 97T^{2} \)
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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

−8.253634940518521836188122133797, −7.54785348936864403667045411077, −6.70379753990914122018578169211, −5.98749365323460614498608291282, −5.20253228390668006084423654025, −4.19581883673472896204984518849, −3.72105258416426060368616365947, −2.57826244612318890835631423731, −1.84611519784816886674064299699, −1.36334794457252965977863678241, 1.36334794457252965977863678241, 1.84611519784816886674064299699, 2.57826244612318890835631423731, 3.72105258416426060368616365947, 4.19581883673472896204984518849, 5.20253228390668006084423654025, 5.98749365323460614498608291282, 6.70379753990914122018578169211, 7.54785348936864403667045411077, 8.253634940518521836188122133797

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