Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.16·3-s − 2.46·5-s − 4.75·7-s − 1.64·9-s − 2.24·11-s + 5.94·13-s − 2.87·15-s + 3.75·17-s + 4.54·19-s − 5.54·21-s + 4.61·23-s + 1.08·25-s − 5.40·27-s + 8.83·29-s + 0.341·31-s − 2.61·33-s + 11.7·35-s + 3.19·37-s + 6.92·39-s + 3.24·41-s − 6.43·43-s + 4.04·45-s − 11.5·47-s + 15.6·49-s + 4.37·51-s − 7.92·53-s + 5.54·55-s + ⋯
L(s)  = 1  + 0.672·3-s − 1.10·5-s − 1.79·7-s − 0.547·9-s − 0.677·11-s + 1.64·13-s − 0.742·15-s + 0.910·17-s + 1.04·19-s − 1.21·21-s + 0.963·23-s + 0.216·25-s − 1.04·27-s + 1.64·29-s + 0.0613·31-s − 0.455·33-s + 1.98·35-s + 0.524·37-s + 1.10·39-s + 0.506·41-s − 0.981·43-s + 0.603·45-s − 1.67·47-s + 2.23·49-s + 0.612·51-s − 1.08·53-s + 0.747·55-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  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 1.16T + 3T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 + 4.75T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 - 5.94T + 13T^{2} \)
17 \( 1 - 3.75T + 17T^{2} \)
19 \( 1 - 4.54T + 19T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 - 8.83T + 29T^{2} \)
31 \( 1 - 0.341T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 3.24T + 41T^{2} \)
43 \( 1 + 6.43T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 + 5.05T + 59T^{2} \)
61 \( 1 + 0.698T + 61T^{2} \)
67 \( 1 + 7.34T + 67T^{2} \)
71 \( 1 + 3.69T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 5.53T + 79T^{2} \)
83 \( 1 + 5.90T + 83T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 - 0.699T + 97T^{2} \)
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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

−7.85893420277358187515460879685, −7.12747936361035189054968050850, −6.28524023588478566305966003923, −5.78384890441402945265381739547, −4.70281254217136152151602164940, −3.63761366689438674794846980300, −3.14947580267079053932742081686, −2.93909826997393703857344158982, −1.14045958051574455883897326198, 0, 1.14045958051574455883897326198, 2.93909826997393703857344158982, 3.14947580267079053932742081686, 3.63761366689438674794846980300, 4.70281254217136152151602164940, 5.78384890441402945265381739547, 6.28524023588478566305966003923, 7.12747936361035189054968050850, 7.85893420277358187515460879685

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