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  − 0.252·3-s − 1.14·5-s − 2.95·7-s − 2.93·9-s + 4.60·11-s − 2.71·13-s + 0.288·15-s + 6.70·17-s − 3.81·19-s + 0.744·21-s + 8.43·23-s − 3.69·25-s + 1.49·27-s − 8.76·29-s + 5.25·31-s − 1.16·33-s + 3.37·35-s + 5.29·37-s + 0.683·39-s − 7.12·41-s + 10.1·43-s + 3.35·45-s − 4.85·47-s + 1.72·49-s − 1.69·51-s + 9.41·53-s − 5.26·55-s + ⋯
L(s)  = 1  − 0.145·3-s − 0.511·5-s − 1.11·7-s − 0.978·9-s + 1.38·11-s − 0.752·13-s + 0.0743·15-s + 1.62·17-s − 0.874·19-s + 0.162·21-s + 1.75·23-s − 0.738·25-s + 0.287·27-s − 1.62·29-s + 0.943·31-s − 0.202·33-s + 0.570·35-s + 0.870·37-s + 0.109·39-s − 1.11·41-s + 1.55·43-s + 0.500·45-s − 0.708·47-s + 0.246·49-s − 0.236·51-s + 1.29·53-s − 0.710·55-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 \( 1 + 0.252T + 3T^{2} \)
5 \( 1 + 1.14T + 5T^{2} \)
7 \( 1 + 2.95T + 7T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 + 2.71T + 13T^{2} \)
17 \( 1 - 6.70T + 17T^{2} \)
19 \( 1 + 3.81T + 19T^{2} \)
23 \( 1 - 8.43T + 23T^{2} \)
29 \( 1 + 8.76T + 29T^{2} \)
31 \( 1 - 5.25T + 31T^{2} \)
37 \( 1 - 5.29T + 37T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 4.85T + 47T^{2} \)
53 \( 1 - 9.41T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 0.810T + 67T^{2} \)
71 \( 1 + 9.99T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 + 5.78T + 79T^{2} \)
83 \( 1 + 2.78T + 83T^{2} \)
89 \( 1 - 5.17T + 89T^{2} \)
97 \( 1 - 0.683T + 97T^{2} \)
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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

−7.60301695879975453989678035137, −7.05068874504608106908869612851, −6.23164265202343822507413145790, −5.76398736269130233908931609495, −4.83679773646260332420319658203, −3.84832062370655932818818095664, −3.33011007515380891949540381213, −2.52525215225977503674750601350, −1.11826978994614781069470030450, 0, 1.11826978994614781069470030450, 2.52525215225977503674750601350, 3.33011007515380891949540381213, 3.84832062370655932818818095664, 4.83679773646260332420319658203, 5.76398736269130233908931609495, 6.23164265202343822507413145790, 7.05068874504608106908869612851, 7.60301695879975453989678035137

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