Properties

Label 2-1008-1.1-c3-0-39
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s + 7·7-s − 40·11-s − 12·13-s + 58·17-s − 26·19-s − 64·23-s − 61·25-s + 62·29-s − 252·31-s + 56·35-s + 26·37-s − 6·41-s − 416·43-s − 396·47-s + 49·49-s + 450·53-s − 320·55-s + 274·59-s − 576·61-s − 96·65-s + 476·67-s − 448·71-s − 158·73-s − 280·77-s + 936·79-s + 530·83-s + ⋯
L(s)  = 1  + 0.715·5-s + 0.377·7-s − 1.09·11-s − 0.256·13-s + 0.827·17-s − 0.313·19-s − 0.580·23-s − 0.487·25-s + 0.397·29-s − 1.46·31-s + 0.270·35-s + 0.115·37-s − 0.0228·41-s − 1.47·43-s − 1.22·47-s + 1/7·49-s + 1.16·53-s − 0.784·55-s + 0.604·59-s − 1.20·61-s − 0.183·65-s + 0.867·67-s − 0.748·71-s − 0.253·73-s − 0.414·77-s + 1.33·79-s + 0.700·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - p T \)
good5 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 + 40 T + p^{3} T^{2} \)
13 \( 1 + 12 T + p^{3} T^{2} \)
17 \( 1 - 58 T + p^{3} T^{2} \)
19 \( 1 + 26 T + p^{3} T^{2} \)
23 \( 1 + 64 T + p^{3} T^{2} \)
29 \( 1 - 62 T + p^{3} T^{2} \)
31 \( 1 + 252 T + p^{3} T^{2} \)
37 \( 1 - 26 T + p^{3} T^{2} \)
41 \( 1 + 6 T + p^{3} T^{2} \)
43 \( 1 + 416 T + p^{3} T^{2} \)
47 \( 1 + 396 T + p^{3} T^{2} \)
53 \( 1 - 450 T + p^{3} T^{2} \)
59 \( 1 - 274 T + p^{3} T^{2} \)
61 \( 1 + 576 T + p^{3} T^{2} \)
67 \( 1 - 476 T + p^{3} T^{2} \)
71 \( 1 + 448 T + p^{3} T^{2} \)
73 \( 1 + 158 T + p^{3} T^{2} \)
79 \( 1 - 936 T + p^{3} T^{2} \)
83 \( 1 - 530 T + p^{3} T^{2} \)
89 \( 1 - 390 T + p^{3} T^{2} \)
97 \( 1 - 214 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.279274704300547730523524788687, −8.216143852122074665633956411205, −7.64375968122612010146055755287, −6.55880109228085385189909556688, −5.56412989895551819632135340069, −5.02873362925692431004492977050, −3.72850842341197016567627414298, −2.53346841122228334214817690594, −1.59868074717443280620767194070, 0, 1.59868074717443280620767194070, 2.53346841122228334214817690594, 3.72850842341197016567627414298, 5.02873362925692431004492977050, 5.56412989895551819632135340069, 6.55880109228085385189909556688, 7.64375968122612010146055755287, 8.216143852122074665633956411205, 9.279274704300547730523524788687

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