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  − 0.753·3-s − 1.16·5-s + 1.19·7-s − 2.43·9-s − 3.93·11-s + 2.90·13-s + 0.878·15-s − 1.93·17-s + 3.04·19-s − 0.897·21-s + 6.33·23-s − 3.64·25-s + 4.09·27-s + 3.20·29-s + 0.203·31-s + 2.96·33-s − 1.38·35-s − 5.22·37-s − 2.18·39-s + 10.5·41-s − 6.50·43-s + 2.83·45-s − 1.53·47-s − 5.58·49-s + 1.46·51-s − 0.0388·53-s + 4.58·55-s + ⋯
L(s)  = 1  − 0.435·3-s − 0.521·5-s + 0.449·7-s − 0.810·9-s − 1.18·11-s + 0.805·13-s + 0.226·15-s − 0.469·17-s + 0.697·19-s − 0.195·21-s + 1.32·23-s − 0.728·25-s + 0.788·27-s + 0.595·29-s + 0.0365·31-s + 0.516·33-s − 0.234·35-s − 0.859·37-s − 0.350·39-s + 1.64·41-s − 0.992·43-s + 0.422·45-s − 0.223·47-s − 0.797·49-s + 0.204·51-s − 0.00533·53-s + 0.618·55-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 \( 1 + 0.753T + 3T^{2} \)
5 \( 1 + 1.16T + 5T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 + 3.93T + 11T^{2} \)
13 \( 1 - 2.90T + 13T^{2} \)
17 \( 1 + 1.93T + 17T^{2} \)
19 \( 1 - 3.04T + 19T^{2} \)
23 \( 1 - 6.33T + 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 - 0.203T + 31T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 6.50T + 43T^{2} \)
47 \( 1 + 1.53T + 47T^{2} \)
53 \( 1 + 0.0388T + 53T^{2} \)
59 \( 1 - 4.00T + 59T^{2} \)
61 \( 1 - 5.53T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 2.94T + 71T^{2} \)
73 \( 1 - 7.90T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + 0.232T + 89T^{2} \)
97 \( 1 - 5.21T + 97T^{2} \)
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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

−7.86893563771292850167417611294, −7.01300509844928704666526206941, −6.29815644394866881153025354145, −5.34127071578539529782022469196, −5.11556428070632798070472697202, −4.07584357690614646632785104969, −3.18282181529445987700095638210, −2.45225199569952313136119447718, −1.13643916508057086361612997605, 0, 1.13643916508057086361612997605, 2.45225199569952313136119447718, 3.18282181529445987700095638210, 4.07584357690614646632785104969, 5.11556428070632798070472697202, 5.34127071578539529782022469196, 6.29815644394866881153025354145, 7.01300509844928704666526206941, 7.86893563771292850167417611294

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