Properties

Label 2-1008-1.1-c5-0-2
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 44·5-s + 49·7-s − 470·11-s − 1.15e3·13-s − 1.20e3·17-s + 2.64e3·19-s − 1.19e3·23-s − 1.18e3·25-s − 3.61e3·29-s − 5.61e3·31-s − 2.15e3·35-s − 6.47e3·37-s − 2.85e3·41-s + 1.34e4·43-s − 1.83e4·47-s + 2.40e3·49-s + 4.37e3·53-s + 2.06e4·55-s + 3.02e4·59-s + 1.95e4·61-s + 5.09e4·65-s − 5.43e4·67-s − 1.07e4·71-s + 3.53e4·73-s − 2.30e4·77-s + 4.99e4·79-s − 2.69e4·83-s + ⋯
L(s)  = 1  − 0.787·5-s + 0.377·7-s − 1.17·11-s − 1.90·13-s − 1.01·17-s + 1.68·19-s − 0.469·23-s − 0.380·25-s − 0.797·29-s − 1.04·31-s − 0.297·35-s − 0.777·37-s − 0.265·41-s + 1.11·43-s − 1.21·47-s + 1/7·49-s + 0.213·53-s + 0.921·55-s + 1.13·59-s + 0.672·61-s + 1.49·65-s − 1.47·67-s − 0.252·71-s + 0.776·73-s − 0.442·77-s + 0.900·79-s − 0.429·83-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5289659038\)
\(L(\frac12)\) \(\approx\) \(0.5289659038\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
good5 \( 1 + 44 T + p^{5} T^{2} \)
11 \( 1 + 470 T + p^{5} T^{2} \)
13 \( 1 + 1158 T + p^{5} T^{2} \)
17 \( 1 + 1204 T + p^{5} T^{2} \)
19 \( 1 - 2644 T + p^{5} T^{2} \)
23 \( 1 + 1190 T + p^{5} T^{2} \)
29 \( 1 + 3614 T + p^{5} T^{2} \)
31 \( 1 + 5616 T + p^{5} T^{2} \)
37 \( 1 + 6478 T + p^{5} T^{2} \)
41 \( 1 + 2856 T + p^{5} T^{2} \)
43 \( 1 - 13492 T + p^{5} T^{2} \)
47 \( 1 + 18372 T + p^{5} T^{2} \)
53 \( 1 - 4374 T + p^{5} T^{2} \)
59 \( 1 - 30248 T + p^{5} T^{2} \)
61 \( 1 - 19542 T + p^{5} T^{2} \)
67 \( 1 + 54328 T + p^{5} T^{2} \)
71 \( 1 + 10730 T + p^{5} T^{2} \)
73 \( 1 - 35374 T + p^{5} T^{2} \)
79 \( 1 - 49956 T + p^{5} T^{2} \)
83 \( 1 + 26948 T + p^{5} T^{2} \)
89 \( 1 + 100776 T + p^{5} T^{2} \)
97 \( 1 - 77134 T + p^{5} T^{2} \)
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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

−9.324959332375718276403075172855, −8.188732080101383254957965741200, −7.51476379054258662339675995611, −7.09401126807709909929651272936, −5.50986278018014588634860450915, −4.98855975162441628537382497760, −3.97751733199079076445625588746, −2.84645599800362099640769574170, −1.93783202122596153422693362925, −0.29814250369677572606020489766, 0.29814250369677572606020489766, 1.93783202122596153422693362925, 2.84645599800362099640769574170, 3.97751733199079076445625588746, 4.98855975162441628537382497760, 5.50986278018014588634860450915, 7.09401126807709909929651272936, 7.51476379054258662339675995611, 8.188732080101383254957965741200, 9.324959332375718276403075172855

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