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  + 2.95·3-s − 2.20·5-s + 2.79·7-s + 5.72·9-s − 1.59·11-s − 3.09·13-s − 6.52·15-s − 6.40·17-s − 3.68·19-s + 8.24·21-s + 3.82·23-s − 0.117·25-s + 8.04·27-s + 1.84·29-s − 9.59·31-s − 4.70·33-s − 6.16·35-s − 9.44·37-s − 9.14·39-s − 9.33·41-s − 2.49·43-s − 12.6·45-s − 2.81·47-s + 0.789·49-s − 18.9·51-s + 1.77·53-s + 3.51·55-s + ⋯
L(s)  = 1  + 1.70·3-s − 0.988·5-s + 1.05·7-s + 1.90·9-s − 0.480·11-s − 0.858·13-s − 1.68·15-s − 1.55·17-s − 0.846·19-s + 1.79·21-s + 0.798·23-s − 0.0235·25-s + 1.54·27-s + 0.343·29-s − 1.72·31-s − 0.818·33-s − 1.04·35-s − 1.55·37-s − 1.46·39-s − 1.45·41-s − 0.381·43-s − 1.88·45-s − 0.410·47-s + 0.112·49-s − 2.64·51-s + 0.243·53-s + 0.474·55-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 \( 1 - 2.95T + 3T^{2} \)
5 \( 1 + 2.20T + 5T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 + 1.59T + 11T^{2} \)
13 \( 1 + 3.09T + 13T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 + 3.68T + 19T^{2} \)
23 \( 1 - 3.82T + 23T^{2} \)
29 \( 1 - 1.84T + 29T^{2} \)
31 \( 1 + 9.59T + 31T^{2} \)
37 \( 1 + 9.44T + 37T^{2} \)
41 \( 1 + 9.33T + 41T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 + 2.81T + 47T^{2} \)
53 \( 1 - 1.77T + 53T^{2} \)
59 \( 1 - 5.74T + 59T^{2} \)
61 \( 1 - 2.55T + 61T^{2} \)
67 \( 1 - 7.92T + 67T^{2} \)
71 \( 1 - 0.0832T + 71T^{2} \)
73 \( 1 + 2.15T + 73T^{2} \)
79 \( 1 - 3.81T + 79T^{2} \)
83 \( 1 + 2.22T + 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−7.905523911453740412004446724334, −7.17847682356080410205053891666, −6.82983539122461240954525812087, −5.20656750305105644380686828401, −4.65491234204823456611339399858, −3.92721907136378165571141873405, −3.26298111748844213434357452544, −2.24761997036934579706047577147, −1.81293191334837924359326119250, 0, 1.81293191334837924359326119250, 2.24761997036934579706047577147, 3.26298111748844213434357452544, 3.92721907136378165571141873405, 4.65491234204823456611339399858, 5.20656750305105644380686828401, 6.82983539122461240954525812087, 7.17847682356080410205053891666, 7.905523911453740412004446724334

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