Properties

Label 2-1008-1.1-c5-0-44
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 23.7·5-s − 49·7-s − 590.·11-s + 505.·13-s + 517.·17-s − 932.·19-s + 3.55e3·23-s − 2.55e3·25-s + 5.39e3·29-s + 8.32e3·31-s + 1.16e3·35-s − 4.93e3·37-s + 5.88e3·41-s − 1.35e4·43-s + 7.35e3·47-s + 2.40e3·49-s − 3.34e3·53-s + 1.40e4·55-s − 4.26e4·59-s + 5.13e4·61-s − 1.20e4·65-s + 3.37e4·67-s + 1.78e4·71-s − 2.06e4·73-s + 2.89e4·77-s − 9.92e4·79-s − 6.12e4·83-s + ⋯
L(s)  = 1  − 0.425·5-s − 0.377·7-s − 1.47·11-s + 0.828·13-s + 0.434·17-s − 0.592·19-s + 1.39·23-s − 0.819·25-s + 1.19·29-s + 1.55·31-s + 0.160·35-s − 0.592·37-s + 0.546·41-s − 1.11·43-s + 0.485·47-s + 0.142·49-s − 0.163·53-s + 0.626·55-s − 1.59·59-s + 1.76·61-s − 0.352·65-s + 0.918·67-s + 0.420·71-s − 0.453·73-s + 0.556·77-s − 1.78·79-s − 0.975·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 49T \)
good5 \( 1 + 23.7T + 3.12e3T^{2} \)
11 \( 1 + 590.T + 1.61e5T^{2} \)
13 \( 1 - 505.T + 3.71e5T^{2} \)
17 \( 1 - 517.T + 1.41e6T^{2} \)
19 \( 1 + 932.T + 2.47e6T^{2} \)
23 \( 1 - 3.55e3T + 6.43e6T^{2} \)
29 \( 1 - 5.39e3T + 2.05e7T^{2} \)
31 \( 1 - 8.32e3T + 2.86e7T^{2} \)
37 \( 1 + 4.93e3T + 6.93e7T^{2} \)
41 \( 1 - 5.88e3T + 1.15e8T^{2} \)
43 \( 1 + 1.35e4T + 1.47e8T^{2} \)
47 \( 1 - 7.35e3T + 2.29e8T^{2} \)
53 \( 1 + 3.34e3T + 4.18e8T^{2} \)
59 \( 1 + 4.26e4T + 7.14e8T^{2} \)
61 \( 1 - 5.13e4T + 8.44e8T^{2} \)
67 \( 1 - 3.37e4T + 1.35e9T^{2} \)
71 \( 1 - 1.78e4T + 1.80e9T^{2} \)
73 \( 1 + 2.06e4T + 2.07e9T^{2} \)
79 \( 1 + 9.92e4T + 3.07e9T^{2} \)
83 \( 1 + 6.12e4T + 3.93e9T^{2} \)
89 \( 1 - 9.01e4T + 5.58e9T^{2} \)
97 \( 1 + 1.55e5T + 8.58e9T^{2} \)
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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

−8.571732695965850252394889825880, −8.139875858640195292508013700829, −7.15730525023945646004831056517, −6.28421686905417603119312002871, −5.33381366131655818897636681513, −4.44148536566129612663522716619, −3.31636713720078972671877368233, −2.54613259518314325278777639878, −1.07826146232014136088977293964, 0, 1.07826146232014136088977293964, 2.54613259518314325278777639878, 3.31636713720078972671877368233, 4.44148536566129612663522716619, 5.33381366131655818897636681513, 6.28421686905417603119312002871, 7.15730525023945646004831056517, 8.139875858640195292508013700829, 8.571732695965850252394889825880

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