Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.83·3-s + 4.26·5-s + 3.89·7-s + 5.01·9-s + 2.11·11-s + 2.93·13-s − 12.0·15-s − 2.84·17-s − 7.25·19-s − 11.0·21-s + 0.341·23-s + 13.1·25-s − 5.71·27-s + 0.780·29-s + 5.53·31-s − 6.00·33-s + 16.6·35-s + 6.82·37-s − 8.31·39-s − 8.86·41-s − 4.55·43-s + 21.3·45-s − 9.00·47-s + 8.20·49-s + 8.05·51-s + 4.97·53-s + 9.03·55-s + ⋯
L(s)  = 1  − 1.63·3-s + 1.90·5-s + 1.47·7-s + 1.67·9-s + 0.638·11-s + 0.814·13-s − 3.11·15-s − 0.689·17-s − 1.66·19-s − 2.40·21-s + 0.0712·23-s + 2.63·25-s − 1.09·27-s + 0.144·29-s + 0.994·31-s − 1.04·33-s + 2.81·35-s + 1.12·37-s − 1.33·39-s − 1.38·41-s − 0.694·43-s + 3.18·45-s − 1.31·47-s + 1.17·49-s + 1.12·51-s + 0.683·53-s + 1.21·55-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  =  0
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.258019599$
$L(\frac12)$  $\approx$  $2.258019599$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 + 2.83T + 3T^{2} \)
5 \( 1 - 4.26T + 5T^{2} \)
7 \( 1 - 3.89T + 7T^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
13 \( 1 - 2.93T + 13T^{2} \)
17 \( 1 + 2.84T + 17T^{2} \)
19 \( 1 + 7.25T + 19T^{2} \)
23 \( 1 - 0.341T + 23T^{2} \)
29 \( 1 - 0.780T + 29T^{2} \)
31 \( 1 - 5.53T + 31T^{2} \)
37 \( 1 - 6.82T + 37T^{2} \)
41 \( 1 + 8.86T + 41T^{2} \)
43 \( 1 + 4.55T + 43T^{2} \)
47 \( 1 + 9.00T + 47T^{2} \)
53 \( 1 - 4.97T + 53T^{2} \)
59 \( 1 - 1.52T + 59T^{2} \)
61 \( 1 - 7.97T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 8.40T + 73T^{2} \)
79 \( 1 + 2.38T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 - 4.05T + 97T^{2} \)
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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

−8.306301332359346106466976483767, −6.87149125625021283854116961875, −6.38783089132247508741484254217, −6.11456658779380964434113829641, −5.07601460527558048573991258101, −4.94853796411318027220087265175, −4.01154838104988148761751028789, −2.30620647605242081279930081730, −1.67756302723900799830886467230, −0.941620008776457648681087815289, 0.941620008776457648681087815289, 1.67756302723900799830886467230, 2.30620647605242081279930081730, 4.01154838104988148761751028789, 4.94853796411318027220087265175, 5.07601460527558048573991258101, 6.11456658779380964434113829641, 6.38783089132247508741484254217, 6.87149125625021283854116961875, 8.306301332359346106466976483767

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