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  + 0.278·3-s + 1.58·5-s − 2.49·7-s − 2.92·9-s + 5.31·11-s − 4.17·13-s + 0.441·15-s − 3.16·17-s + 5.18·19-s − 0.693·21-s − 2.27·23-s − 2.48·25-s − 1.64·27-s + 4.70·29-s − 0.466·31-s + 1.47·33-s − 3.95·35-s + 4.71·37-s − 1.16·39-s + 4.53·41-s + 3.13·43-s − 4.63·45-s − 1.59·47-s − 0.791·49-s − 0.879·51-s − 3.67·53-s + 8.43·55-s + ⋯
L(s)  = 1  + 0.160·3-s + 0.709·5-s − 0.941·7-s − 0.974·9-s + 1.60·11-s − 1.15·13-s + 0.113·15-s − 0.766·17-s + 1.18·19-s − 0.151·21-s − 0.473·23-s − 0.496·25-s − 0.317·27-s + 0.874·29-s − 0.0837·31-s + 0.257·33-s − 0.668·35-s + 0.775·37-s − 0.186·39-s + 0.708·41-s + 0.477·43-s − 0.691·45-s − 0.233·47-s − 0.113·49-s − 0.123·51-s − 0.505·53-s + 1.13·55-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 \( 1 - 0.278T + 3T^{2} \)
5 \( 1 - 1.58T + 5T^{2} \)
7 \( 1 + 2.49T + 7T^{2} \)
11 \( 1 - 5.31T + 11T^{2} \)
13 \( 1 + 4.17T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 - 5.18T + 19T^{2} \)
23 \( 1 + 2.27T + 23T^{2} \)
29 \( 1 - 4.70T + 29T^{2} \)
31 \( 1 + 0.466T + 31T^{2} \)
37 \( 1 - 4.71T + 37T^{2} \)
41 \( 1 - 4.53T + 41T^{2} \)
43 \( 1 - 3.13T + 43T^{2} \)
47 \( 1 + 1.59T + 47T^{2} \)
53 \( 1 + 3.67T + 53T^{2} \)
59 \( 1 + 3.30T + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 5.02T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 - 3.89T + 97T^{2} \)
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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

−7.64175709224360638384974920132, −6.93986537877005432006336450024, −6.15862458719317994568556532433, −5.88163533091232481369608172257, −4.81940659860078789406026519169, −4.00742475092763398356021744578, −3.06255348623404932086407995962, −2.48549933870109822041902682040, −1.37602980463858779022753745080, 0, 1.37602980463858779022753745080, 2.48549933870109822041902682040, 3.06255348623404932086407995962, 4.00742475092763398356021744578, 4.81940659860078789406026519169, 5.88163533091232481369608172257, 6.15862458719317994568556532433, 6.93986537877005432006336450024, 7.64175709224360638384974920132

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