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  + 3.42·3-s + 1.19·5-s − 1.06·7-s + 8.73·9-s + 0.730·11-s + 5.49·13-s + 4.08·15-s + 0.933·17-s + 5.09·19-s − 3.63·21-s − 6.73·23-s − 3.58·25-s + 19.6·27-s + 5.95·29-s + 7.12·31-s + 2.50·33-s − 1.26·35-s + 7.45·37-s + 18.8·39-s − 11.8·41-s − 12.0·43-s + 10.4·45-s − 11.0·47-s − 5.87·49-s + 3.19·51-s − 7.95·53-s + 0.870·55-s + ⋯
L(s)  = 1  + 1.97·3-s + 0.532·5-s − 0.400·7-s + 2.91·9-s + 0.220·11-s + 1.52·13-s + 1.05·15-s + 0.226·17-s + 1.16·19-s − 0.792·21-s − 1.40·23-s − 0.716·25-s + 3.78·27-s + 1.10·29-s + 1.27·31-s + 0.435·33-s − 0.213·35-s + 1.22·37-s + 3.01·39-s − 1.85·41-s − 1.83·43-s + 1.55·45-s − 1.61·47-s − 0.839·49-s + 0.447·51-s − 1.09·53-s + 0.117·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.447262838$
$L(\frac12)$  $\approx$  $5.447262838$
$L(\frac{3}{2})$   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 \( 1 - 3.42T + 3T^{2} \)
5 \( 1 - 1.19T + 5T^{2} \)
7 \( 1 + 1.06T + 7T^{2} \)
11 \( 1 - 0.730T + 11T^{2} \)
13 \( 1 - 5.49T + 13T^{2} \)
17 \( 1 - 0.933T + 17T^{2} \)
19 \( 1 - 5.09T + 19T^{2} \)
23 \( 1 + 6.73T + 23T^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 - 7.12T + 31T^{2} \)
37 \( 1 - 7.45T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 7.95T + 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 0.0992T + 67T^{2} \)
71 \( 1 + 0.301T + 71T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 - 0.440T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 5.62T + 89T^{2} \)
97 \( 1 - 2.36T + 97T^{2} \)
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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

−8.271494860291854743756696427407, −7.70244562444323186196964484010, −6.52841856052154988122858115140, −6.36820370272330466548571956431, −5.02763442848833144357577051744, −4.13792283542897478167010776906, −3.36473528685962110368444472153, −3.01510179532937115545081123924, −1.86781538172883336085712541321, −1.29557962209679009276266506603, 1.29557962209679009276266506603, 1.86781538172883336085712541321, 3.01510179532937115545081123924, 3.36473528685962110368444472153, 4.13792283542897478167010776906, 5.02763442848833144357577051744, 6.36820370272330466548571956431, 6.52841856052154988122858115140, 7.70244562444323186196964484010, 8.271494860291854743756696427407

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