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  − 1.75·3-s + 1.89·5-s − 3.72·7-s + 0.0728·9-s + 5.05·11-s + 3.05·13-s − 3.32·15-s + 0.674·17-s − 1.01·19-s + 6.53·21-s − 4.16·23-s − 1.40·25-s + 5.13·27-s − 6.08·29-s + 1.11·31-s − 8.86·33-s − 7.07·35-s − 8.11·37-s − 5.35·39-s + 9.59·41-s − 10.0·43-s + 0.138·45-s − 2.58·47-s + 6.91·49-s − 1.18·51-s + 8.96·53-s + 9.59·55-s + ⋯
L(s)  = 1  − 1.01·3-s + 0.848·5-s − 1.40·7-s + 0.0242·9-s + 1.52·11-s + 0.847·13-s − 0.858·15-s + 0.163·17-s − 0.233·19-s + 1.42·21-s − 0.868·23-s − 0.280·25-s + 0.987·27-s − 1.12·29-s + 0.200·31-s − 1.54·33-s − 1.19·35-s − 1.33·37-s − 0.857·39-s + 1.49·41-s − 1.53·43-s + 0.0205·45-s − 0.377·47-s + 0.987·49-s − 0.165·51-s + 1.23·53-s + 1.29·55-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 \( 1 + 1.75T + 3T^{2} \)
5 \( 1 - 1.89T + 5T^{2} \)
7 \( 1 + 3.72T + 7T^{2} \)
11 \( 1 - 5.05T + 11T^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 - 0.674T + 17T^{2} \)
19 \( 1 + 1.01T + 19T^{2} \)
23 \( 1 + 4.16T + 23T^{2} \)
29 \( 1 + 6.08T + 29T^{2} \)
31 \( 1 - 1.11T + 31T^{2} \)
37 \( 1 + 8.11T + 37T^{2} \)
41 \( 1 - 9.59T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 8.96T + 53T^{2} \)
59 \( 1 + 4.15T + 59T^{2} \)
61 \( 1 - 7.74T + 61T^{2} \)
67 \( 1 + 4.80T + 67T^{2} \)
71 \( 1 + 8.83T + 71T^{2} \)
73 \( 1 + 0.804T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 0.249T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 6.87T + 97T^{2} \)
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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

−7.50509681244885420685616881079, −6.59075906475965765304932137198, −6.22101481822865624731504746431, −5.91991078660050851080636183792, −5.05975750403468523235774058923, −3.87887900129805060993849695462, −3.46250695594827181343675619615, −2.19199053732584294834489196599, −1.19294278060123591951712247758, 0, 1.19294278060123591951712247758, 2.19199053732584294834489196599, 3.46250695594827181343675619615, 3.87887900129805060993849695462, 5.05975750403468523235774058923, 5.91991078660050851080636183792, 6.22101481822865624731504746431, 6.59075906475965765304932137198, 7.50509681244885420685616881079

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