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  + 2.35·3-s − 3.32·5-s + 1.00·7-s + 2.56·9-s − 0.281·11-s + 0.731·13-s − 7.83·15-s + 4.74·17-s − 5.22·19-s + 2.36·21-s − 6.35·23-s + 6.03·25-s − 1.02·27-s + 2.12·29-s + 1.17·31-s − 0.664·33-s − 3.32·35-s + 1.39·37-s + 1.72·39-s − 2.17·41-s − 11.2·43-s − 8.52·45-s + 2.59·47-s − 5.99·49-s + 11.2·51-s + 9.36·53-s + 0.935·55-s + ⋯
L(s)  = 1  + 1.36·3-s − 1.48·5-s + 0.378·7-s + 0.855·9-s − 0.0848·11-s + 0.202·13-s − 2.02·15-s + 1.15·17-s − 1.19·19-s + 0.515·21-s − 1.32·23-s + 1.20·25-s − 0.196·27-s + 0.394·29-s + 0.210·31-s − 0.115·33-s − 0.562·35-s + 0.230·37-s + 0.276·39-s − 0.340·41-s − 1.71·43-s − 1.27·45-s + 0.377·47-s − 0.856·49-s + 1.56·51-s + 1.28·53-s + 0.126·55-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 \( 1 - 2.35T + 3T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
7 \( 1 - 1.00T + 7T^{2} \)
11 \( 1 + 0.281T + 11T^{2} \)
13 \( 1 - 0.731T + 13T^{2} \)
17 \( 1 - 4.74T + 17T^{2} \)
19 \( 1 + 5.22T + 19T^{2} \)
23 \( 1 + 6.35T + 23T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 - 1.39T + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 2.59T + 47T^{2} \)
53 \( 1 - 9.36T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 2.15T + 67T^{2} \)
71 \( 1 - 4.52T + 71T^{2} \)
73 \( 1 - 3.16T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 2.06T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−7.976071172573929816494504711376, −7.34716122128180009188429117368, −6.53176507087284386785955633894, −5.49305226623947589534784628129, −4.45388498076034497421458588842, −3.91269385997177037814726473857, −3.33196042194176676842389509876, −2.50108156780215802945501460728, −1.48389686407987205519219669228, 0, 1.48389686407987205519219669228, 2.50108156780215802945501460728, 3.33196042194176676842389509876, 3.91269385997177037814726473857, 4.45388498076034497421458588842, 5.49305226623947589534784628129, 6.53176507087284386785955633894, 7.34716122128180009188429117368, 7.976071172573929816494504711376

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