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  + 0.343·3-s − 0.515·5-s − 3.09·7-s − 2.88·9-s + 3.36·11-s + 6.89·13-s − 0.177·15-s − 2.74·17-s + 5.90·19-s − 1.06·21-s + 0.434·23-s − 4.73·25-s − 2.02·27-s + 0.715·29-s − 0.476·31-s + 1.15·33-s + 1.59·35-s − 10.3·37-s + 2.36·39-s + 5.09·41-s − 2.97·43-s + 1.48·45-s + 5.04·47-s + 2.59·49-s − 0.942·51-s + 8.94·53-s − 1.73·55-s + ⋯
L(s)  = 1  + 0.198·3-s − 0.230·5-s − 1.17·7-s − 0.960·9-s + 1.01·11-s + 1.91·13-s − 0.0457·15-s − 0.665·17-s + 1.35·19-s − 0.232·21-s + 0.0905·23-s − 0.946·25-s − 0.388·27-s + 0.132·29-s − 0.0855·31-s + 0.201·33-s + 0.270·35-s − 1.70·37-s + 0.378·39-s + 0.795·41-s − 0.453·43-s + 0.221·45-s + 0.735·47-s + 0.371·49-s − 0.131·51-s + 1.22·53-s − 0.234·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.665487671$
$L(\frac12)$  $\approx$  $1.665487671$
$L(\frac{3}{2})$   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 \( 1 - 0.343T + 3T^{2} \)
5 \( 1 + 0.515T + 5T^{2} \)
7 \( 1 + 3.09T + 7T^{2} \)
11 \( 1 - 3.36T + 11T^{2} \)
13 \( 1 - 6.89T + 13T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 - 5.90T + 19T^{2} \)
23 \( 1 - 0.434T + 23T^{2} \)
29 \( 1 - 0.715T + 29T^{2} \)
31 \( 1 + 0.476T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 5.09T + 41T^{2} \)
43 \( 1 + 2.97T + 43T^{2} \)
47 \( 1 - 5.04T + 47T^{2} \)
53 \( 1 - 8.94T + 53T^{2} \)
59 \( 1 + 9.71T + 59T^{2} \)
61 \( 1 + 9.74T + 61T^{2} \)
67 \( 1 - 7.35T + 67T^{2} \)
71 \( 1 + 1.40T + 71T^{2} \)
73 \( 1 - 16.7T + 73T^{2} \)
79 \( 1 + 5.53T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 0.906T + 89T^{2} \)
97 \( 1 - 0.281T + 97T^{2} \)
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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

−8.205071657977816510350193737395, −7.31725273283589368635597340737, −6.53571447213694144368759024716, −6.05840524559849460389031412671, −5.43087482503557925735847213697, −4.16080374665221443703205407261, −3.49511819606247392282231166117, −3.11673999378712411438525886396, −1.81911570254155929804823052157, −0.67363158011824553258098581111, 0.67363158011824553258098581111, 1.81911570254155929804823052157, 3.11673999378712411438525886396, 3.49511819606247392282231166117, 4.16080374665221443703205407261, 5.43087482503557925735847213697, 6.05840524559849460389031412671, 6.53571447213694144368759024716, 7.31725273283589368635597340737, 8.205071657977816510350193737395

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