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  − 0.486·3-s + 3.55·5-s + 0.471·7-s − 2.76·9-s − 0.531·11-s + 4.46·13-s − 1.72·15-s − 4.83·17-s − 7.53·19-s − 0.229·21-s − 6.36·23-s + 7.64·25-s + 2.80·27-s − 0.479·29-s + 2.79·31-s + 0.258·33-s + 1.67·35-s + 10.7·37-s − 2.17·39-s − 0.506·41-s − 10.2·43-s − 9.82·45-s − 0.860·47-s − 6.77·49-s + 2.35·51-s − 6.48·53-s − 1.88·55-s + ⋯
L(s)  = 1  − 0.280·3-s + 1.59·5-s + 0.178·7-s − 0.921·9-s − 0.160·11-s + 1.23·13-s − 0.446·15-s − 1.17·17-s − 1.72·19-s − 0.0500·21-s − 1.32·23-s + 1.52·25-s + 0.539·27-s − 0.0890·29-s + 0.501·31-s + 0.0449·33-s + 0.283·35-s + 1.76·37-s − 0.347·39-s − 0.0790·41-s − 1.55·43-s − 1.46·45-s − 0.125·47-s − 0.968·49-s + 0.329·51-s − 0.891·53-s − 0.254·55-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 \( 1 + 0.486T + 3T^{2} \)
5 \( 1 - 3.55T + 5T^{2} \)
7 \( 1 - 0.471T + 7T^{2} \)
11 \( 1 + 0.531T + 11T^{2} \)
13 \( 1 - 4.46T + 13T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 + 7.53T + 19T^{2} \)
23 \( 1 + 6.36T + 23T^{2} \)
29 \( 1 + 0.479T + 29T^{2} \)
31 \( 1 - 2.79T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 0.506T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 0.860T + 47T^{2} \)
53 \( 1 + 6.48T + 53T^{2} \)
59 \( 1 + 3.90T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 + 9.72T + 67T^{2} \)
71 \( 1 + 7.91T + 71T^{2} \)
73 \( 1 - 4.56T + 73T^{2} \)
79 \( 1 - 2.55T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−7.958610401281914468745661563084, −6.45174937078603446579638320940, −6.32933761744253219124613120645, −5.86417170589453114322390841900, −4.89441383570114807608993806120, −4.21976098533947441253186066398, −3.01885966383472986488512004825, −2.19625144199057162574341632591, −1.55060399552044853317605383267, 0, 1.55060399552044853317605383267, 2.19625144199057162574341632591, 3.01885966383472986488512004825, 4.21976098533947441253186066398, 4.89441383570114807608993806120, 5.86417170589453114322390841900, 6.32933761744253219124613120645, 6.45174937078603446579638320940, 7.958610401281914468745661563084

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