Properties

Label 2-1008-1.1-c5-0-13
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  + 34·5-s + 49·7-s − 332·11-s − 1.02e3·13-s − 922·17-s − 452·19-s − 3.77e3·23-s − 1.96e3·25-s − 1.16e3·29-s + 9.79e3·31-s + 1.66e3·35-s + 2.39e3·37-s + 7.23e3·41-s − 4.65e3·43-s + 2.46e4·47-s + 2.40e3·49-s − 1.11e3·53-s − 1.12e4·55-s + 4.68e4·59-s − 9.76e3·61-s − 3.48e4·65-s + 2.62e4·67-s + 6.54e4·71-s − 5.60e3·73-s − 1.62e4·77-s + 9.84e3·79-s + 6.11e4·83-s + ⋯
L(s)  = 1  + 0.608·5-s + 0.377·7-s − 0.827·11-s − 1.68·13-s − 0.773·17-s − 0.287·19-s − 1.48·23-s − 0.630·25-s − 0.257·29-s + 1.83·31-s + 0.229·35-s + 0.287·37-s + 0.671·41-s − 0.383·43-s + 1.62·47-s + 1/7·49-s − 0.0542·53-s − 0.503·55-s + 1.75·59-s − 0.335·61-s − 1.02·65-s + 0.714·67-s + 1.54·71-s − 0.123·73-s − 0.312·77-s + 0.177·79-s + 0.973·83-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

\(L(3)\) \(\approx\) \(1.669298985\)
\(L(\frac12)\) \(\approx\) \(1.669298985\)
\(L(\frac{7}{2})\) 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 \( 1 - 34 T + p^{5} T^{2} \)
11 \( 1 + 332 T + p^{5} T^{2} \)
13 \( 1 + 1026 T + p^{5} T^{2} \)
17 \( 1 + 922 T + p^{5} T^{2} \)
19 \( 1 + 452 T + p^{5} T^{2} \)
23 \( 1 + 3776 T + p^{5} T^{2} \)
29 \( 1 + 1166 T + p^{5} T^{2} \)
31 \( 1 - 9792 T + p^{5} T^{2} \)
37 \( 1 - 2390 T + p^{5} T^{2} \)
41 \( 1 - 7230 T + p^{5} T^{2} \)
43 \( 1 + 4652 T + p^{5} T^{2} \)
47 \( 1 - 24672 T + p^{5} T^{2} \)
53 \( 1 + 1110 T + p^{5} T^{2} \)
59 \( 1 - 46892 T + p^{5} T^{2} \)
61 \( 1 + 9762 T + p^{5} T^{2} \)
67 \( 1 - 26252 T + p^{5} T^{2} \)
71 \( 1 - 65440 T + p^{5} T^{2} \)
73 \( 1 + 5606 T + p^{5} T^{2} \)
79 \( 1 - 9840 T + p^{5} T^{2} \)
83 \( 1 - 61108 T + p^{5} T^{2} \)
89 \( 1 - 62958 T + p^{5} T^{2} \)
97 \( 1 + 37838 T + p^{5} T^{2} \)
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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

−9.422195607228692330767757295828, −8.262704451759666925533190405472, −7.67871755281125688997518704110, −6.68163179181065651588808875042, −5.74685155071270679210591233614, −4.93163837034198611140546186415, −4.09834808924713531531483434208, −2.48273052121188109874553542845, −2.14616705782101446293335050658, −0.53140040153519630338046294382, 0.53140040153519630338046294382, 2.14616705782101446293335050658, 2.48273052121188109874553542845, 4.09834808924713531531483434208, 4.93163837034198611140546186415, 5.74685155071270679210591233614, 6.68163179181065651588808875042, 7.67871755281125688997518704110, 8.262704451759666925533190405472, 9.422195607228692330767757295828

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