Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 3·9-s − 13-s − 6·17-s + 3·19-s − 9·23-s − 25-s − 10·29-s − 8·31-s + 8·35-s + 11·37-s − 4·43-s + 6·45-s − 3·47-s + 9·49-s − 53-s − 13·59-s + 3·61-s + 12·63-s + 2·65-s + 8·67-s − 8·71-s − 4·73-s + 6·79-s + 9·81-s + 6·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 9-s − 0.277·13-s − 1.45·17-s + 0.688·19-s − 1.87·23-s − 1/5·25-s − 1.85·29-s − 1.43·31-s + 1.35·35-s + 1.80·37-s − 0.609·43-s + 0.894·45-s − 0.437·47-s + 9/7·49-s − 0.137·53-s − 1.69·59-s + 0.384·61-s + 1.51·63-s + 0.248·65-s + 0.977·67-s − 0.949·71-s − 0.468·73-s + 0.675·79-s + 81-s + 0.658·83-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  =  2
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 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−7.52542124636976209696731698498, −6.56870323779876583978884872592, −6.05066158186841973885179566530, −5.34614084489609530249489198255, −4.14580059041744850412142656372, −3.70682191769348489151611267627, −2.90206741182725093286474673544, −2.02655141313196498091509295261, 0, 0, 2.02655141313196498091509295261, 2.90206741182725093286474673544, 3.70682191769348489151611267627, 4.14580059041744850412142656372, 5.34614084489609530249489198255, 6.05066158186841973885179566530, 6.56870323779876583978884872592, 7.52542124636976209696731698498

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