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  − 2.25·3-s + 3.53·5-s + 1.19·7-s + 2.06·9-s − 4.93·11-s − 4.24·13-s − 7.95·15-s − 0.933·17-s + 5.44·19-s − 2.67·21-s − 8.71·23-s + 7.47·25-s + 2.10·27-s + 8.13·29-s + 5.52·31-s + 11.1·33-s + 4.20·35-s + 3.19·37-s + 9.54·39-s + 11.8·41-s − 9.73·43-s + 7.30·45-s − 10.8·47-s − 5.58·49-s + 2.10·51-s − 3.26·53-s − 17.4·55-s + ⋯
L(s)  = 1  − 1.29·3-s + 1.57·5-s + 0.449·7-s + 0.688·9-s − 1.48·11-s − 1.17·13-s − 2.05·15-s − 0.226·17-s + 1.24·19-s − 0.584·21-s − 1.81·23-s + 1.49·25-s + 0.404·27-s + 1.51·29-s + 0.992·31-s + 1.93·33-s + 0.710·35-s + 0.525·37-s + 1.52·39-s + 1.85·41-s − 1.48·43-s + 1.08·45-s − 1.58·47-s − 0.797·49-s + 0.294·51-s − 0.449·53-s − 2.35·55-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 \( 1 + 2.25T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 + 4.93T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 0.933T + 17T^{2} \)
19 \( 1 - 5.44T + 19T^{2} \)
23 \( 1 + 8.71T + 23T^{2} \)
29 \( 1 - 8.13T + 29T^{2} \)
31 \( 1 - 5.52T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 9.73T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 3.26T + 53T^{2} \)
59 \( 1 - 1.18T + 59T^{2} \)
61 \( 1 + 1.59T + 61T^{2} \)
67 \( 1 + 2.80T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 - 7.27T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 2.87T + 89T^{2} \)
97 \( 1 + 4.60T + 97T^{2} \)
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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

−7.79058227282698365898396863483, −6.69320213273672289466544124874, −6.20628869163903044647115674975, −5.50641857980538578524181483151, −5.04338976177392749783212779477, −4.58950213684916665776419000204, −2.87730307117529492364632671193, −2.31179363370600287484719759769, −1.25423428595900978735453049610, 0, 1.25423428595900978735453049610, 2.31179363370600287484719759769, 2.87730307117529492364632671193, 4.58950213684916665776419000204, 5.04338976177392749783212779477, 5.50641857980538578524181483151, 6.20628869163903044647115674975, 6.69320213273672289466544124874, 7.79058227282698365898396863483

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