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  + 1.83·3-s + 4.41·5-s + 3.19·7-s + 0.380·9-s − 5.76·11-s + 6.18·13-s + 8.11·15-s + 2.08·17-s + 2.76·19-s + 5.86·21-s + 0.206·23-s + 14.4·25-s − 4.81·27-s + 3.05·29-s + 5.45·31-s − 10.6·33-s + 14.0·35-s − 5.94·37-s + 11.3·39-s − 4.72·41-s − 1.65·43-s + 1.67·45-s − 6.83·47-s + 3.18·49-s + 3.84·51-s + 8.67·53-s − 25.4·55-s + ⋯
L(s)  = 1  + 1.06·3-s + 1.97·5-s + 1.20·7-s + 0.126·9-s − 1.73·11-s + 1.71·13-s + 2.09·15-s + 0.506·17-s + 0.634·19-s + 1.28·21-s + 0.0430·23-s + 2.89·25-s − 0.926·27-s + 0.567·29-s + 0.978·31-s − 1.84·33-s + 2.38·35-s − 0.976·37-s + 1.82·39-s − 0.737·41-s − 0.252·43-s + 0.250·45-s − 0.997·47-s + 0.454·49-s + 0.537·51-s + 1.19·53-s − 3.43·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.149202932$
$L(\frac12)$  $\approx$  $5.149202932$
$L(\frac{3}{2})$   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 \( 1 - 1.83T + 3T^{2} \)
5 \( 1 - 4.41T + 5T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 + 5.76T + 11T^{2} \)
13 \( 1 - 6.18T + 13T^{2} \)
17 \( 1 - 2.08T + 17T^{2} \)
19 \( 1 - 2.76T + 19T^{2} \)
23 \( 1 - 0.206T + 23T^{2} \)
29 \( 1 - 3.05T + 29T^{2} \)
31 \( 1 - 5.45T + 31T^{2} \)
37 \( 1 + 5.94T + 37T^{2} \)
41 \( 1 + 4.72T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 + 6.83T + 47T^{2} \)
53 \( 1 - 8.67T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 6.58T + 61T^{2} \)
67 \( 1 + 7.21T + 67T^{2} \)
71 \( 1 + 16.6T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 9.03T + 79T^{2} \)
83 \( 1 - 6.69T + 83T^{2} \)
89 \( 1 + 7.60T + 89T^{2} \)
97 \( 1 + 0.349T + 97T^{2} \)
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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

−8.247223599107834751142517416835, −7.64815142084785618516701294901, −6.57958232077184017009086657023, −5.75788846923420100794125532247, −5.34925430827826184443825132238, −4.63413844977132986562397336282, −3.21778101515057111703345797127, −2.77875782837578826189091785547, −1.84757614094903824549893937374, −1.31917738082463226299797999347, 1.31917738082463226299797999347, 1.84757614094903824549893937374, 2.77875782837578826189091785547, 3.21778101515057111703345797127, 4.63413844977132986562397336282, 5.34925430827826184443825132238, 5.75788846923420100794125532247, 6.57958232077184017009086657023, 7.64815142084785618516701294901, 8.247223599107834751142517416835

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