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  + 2.93·3-s + 3.50·5-s + 3.59·7-s + 5.62·9-s + 2.69·11-s − 2.54·13-s + 10.2·15-s + 2.23·17-s − 7.67·19-s + 10.5·21-s − 1.01·23-s + 7.28·25-s + 7.70·27-s + 5.38·29-s − 2.95·31-s + 7.91·33-s + 12.6·35-s − 2.73·37-s − 7.47·39-s + 3.58·41-s − 11.3·43-s + 19.7·45-s − 1.21·47-s + 5.93·49-s + 6.55·51-s − 11.6·53-s + 9.44·55-s + ⋯
L(s)  = 1  + 1.69·3-s + 1.56·5-s + 1.35·7-s + 1.87·9-s + 0.812·11-s − 0.705·13-s + 2.65·15-s + 0.541·17-s − 1.76·19-s + 2.30·21-s − 0.211·23-s + 1.45·25-s + 1.48·27-s + 0.999·29-s − 0.530·31-s + 1.37·33-s + 2.13·35-s − 0.448·37-s − 1.19·39-s + 0.560·41-s − 1.73·43-s + 2.93·45-s − 0.176·47-s + 0.847·49-s + 0.917·51-s − 1.60·53-s + 1.27·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.258141629$
$L(\frac12)$  $\approx$  $6.258141629$
$L(\frac{3}{2})$   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 \( 1 - 2.93T + 3T^{2} \)
5 \( 1 - 3.50T + 5T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
11 \( 1 - 2.69T + 11T^{2} \)
13 \( 1 + 2.54T + 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 + 7.67T + 19T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 - 5.38T + 29T^{2} \)
31 \( 1 + 2.95T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 - 3.58T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 1.21T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 + 1.42T + 61T^{2} \)
67 \( 1 + 1.46T + 67T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 7.93T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 7.71T + 97T^{2} \)
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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

−8.172894512812615024879192696229, −7.62502588808510787723607990318, −6.69329724529006354600094259888, −6.08415287909074536551061740564, −4.95612345354179982240830576025, −4.51097320606058470861630191013, −3.47914606601648521898144606322, −2.55099760202724425881369513936, −1.86101859866089130792548650346, −1.51505297715019024927193510779, 1.51505297715019024927193510779, 1.86101859866089130792548650346, 2.55099760202724425881369513936, 3.47914606601648521898144606322, 4.51097320606058470861630191013, 4.95612345354179982240830576025, 6.08415287909074536551061740564, 6.69329724529006354600094259888, 7.62502588808510787723607990318, 8.172894512812615024879192696229

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