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  + 1.87·3-s − 1.46·5-s − 0.498·7-s + 0.528·9-s + 0.602·11-s + 3.48·13-s − 2.75·15-s − 0.975·17-s + 1.95·19-s − 0.935·21-s − 6.22·23-s − 2.85·25-s − 4.64·27-s − 0.238·29-s − 1.95·31-s + 1.13·33-s + 0.729·35-s + 0.114·37-s + 6.53·39-s − 2.20·41-s − 4.53·43-s − 0.773·45-s + 4.00·47-s − 6.75·49-s − 1.83·51-s + 7.59·53-s − 0.882·55-s + ⋯
L(s)  = 1  + 1.08·3-s − 0.655·5-s − 0.188·7-s + 0.176·9-s + 0.181·11-s + 0.965·13-s − 0.710·15-s − 0.236·17-s + 0.449·19-s − 0.204·21-s − 1.29·23-s − 0.570·25-s − 0.893·27-s − 0.0443·29-s − 0.350·31-s + 0.197·33-s + 0.123·35-s + 0.0188·37-s + 1.04·39-s − 0.344·41-s − 0.692·43-s − 0.115·45-s + 0.583·47-s − 0.964·49-s − 0.256·51-s + 1.04·53-s − 0.118·55-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 \( 1 - 1.87T + 3T^{2} \)
5 \( 1 + 1.46T + 5T^{2} \)
7 \( 1 + 0.498T + 7T^{2} \)
11 \( 1 - 0.602T + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 0.975T + 17T^{2} \)
19 \( 1 - 1.95T + 19T^{2} \)
23 \( 1 + 6.22T + 23T^{2} \)
29 \( 1 + 0.238T + 29T^{2} \)
31 \( 1 + 1.95T + 31T^{2} \)
37 \( 1 - 0.114T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 + 4.53T + 43T^{2} \)
47 \( 1 - 4.00T + 47T^{2} \)
53 \( 1 - 7.59T + 53T^{2} \)
59 \( 1 + 2.21T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 9.31T + 67T^{2} \)
71 \( 1 + 4.58T + 71T^{2} \)
73 \( 1 - 6.26T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 - 2.87T + 83T^{2} \)
89 \( 1 - 8.49T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−7.906113515598254559508579651791, −7.25895828912668901344605016399, −6.30382519076843292688600747822, −5.71113065349376549389861331049, −4.58902120386200185313477137355, −3.72752077083737961953524215631, −3.43592296593347209380840987927, −2.41436700487973378112685069916, −1.50834962042503895040323537312, 0, 1.50834962042503895040323537312, 2.41436700487973378112685069916, 3.43592296593347209380840987927, 3.72752077083737961953524215631, 4.58902120386200185313477137355, 5.71113065349376549389861331049, 6.30382519076843292688600747822, 7.25895828912668901344605016399, 7.906113515598254559508579651791

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