Properties

Label 2-1008-1.1-c5-0-74
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·5-s − 49·7-s − 624·11-s − 708·13-s − 934·17-s − 1.85e3·19-s − 1.12e3·23-s − 2.10e3·25-s + 1.17e3·29-s − 2.90e3·31-s + 1.56e3·35-s − 1.24e4·37-s − 2.66e3·41-s + 7.14e3·43-s − 7.46e3·47-s + 2.40e3·49-s + 2.72e4·53-s + 1.99e4·55-s + 2.49e3·59-s − 1.10e4·61-s + 2.26e4·65-s − 3.97e4·67-s − 6.98e4·71-s + 1.64e4·73-s + 3.05e4·77-s − 7.83e4·79-s + 1.09e5·83-s + ⋯
L(s)  = 1  − 0.572·5-s − 0.377·7-s − 1.55·11-s − 1.16·13-s − 0.783·17-s − 1.18·19-s − 0.441·23-s − 0.672·25-s + 0.259·29-s − 0.543·31-s + 0.216·35-s − 1.49·37-s − 0.247·41-s + 0.589·43-s − 0.493·47-s + 1/7·49-s + 1.33·53-s + 0.890·55-s + 0.0931·59-s − 0.381·61-s + 0.665·65-s − 1.08·67-s − 1.64·71-s + 0.361·73-s + 0.587·77-s − 1.41·79-s + 1.74·83-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: \(2\)
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 \( 1 + p^{2} T \)
good5 \( 1 + 32 T + p^{5} T^{2} \)
11 \( 1 + 624 T + p^{5} T^{2} \)
13 \( 1 + 708 T + p^{5} T^{2} \)
17 \( 1 + 934 T + p^{5} T^{2} \)
19 \( 1 + 1858 T + p^{5} T^{2} \)
23 \( 1 + 1120 T + p^{5} T^{2} \)
29 \( 1 - 1174 T + p^{5} T^{2} \)
31 \( 1 + 2908 T + p^{5} T^{2} \)
37 \( 1 + 12462 T + p^{5} T^{2} \)
41 \( 1 + 2662 T + p^{5} T^{2} \)
43 \( 1 - 7144 T + p^{5} T^{2} \)
47 \( 1 + 7468 T + p^{5} T^{2} \)
53 \( 1 - 27274 T + p^{5} T^{2} \)
59 \( 1 - 2490 T + p^{5} T^{2} \)
61 \( 1 + 11096 T + p^{5} T^{2} \)
67 \( 1 + 39756 T + p^{5} T^{2} \)
71 \( 1 + 69888 T + p^{5} T^{2} \)
73 \( 1 - 16450 T + p^{5} T^{2} \)
79 \( 1 + 78376 T + p^{5} T^{2} \)
83 \( 1 - 109818 T + p^{5} T^{2} \)
89 \( 1 - 56966 T + p^{5} T^{2} \)
97 \( 1 + 115946 T + p^{5} T^{2} \)
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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

−8.318087810712769332036019634096, −7.57796166916851333109689744262, −6.85606205214762704111567081669, −5.74502926540283300462317165451, −4.85209182927646900920657177310, −3.98992189181258745212695278631, −2.79920625373463053088291364502, −2.01946491044733083989285518416, 0, 0, 2.01946491044733083989285518416, 2.79920625373463053088291364502, 3.98992189181258745212695278631, 4.85209182927646900920657177310, 5.74502926540283300462317165451, 6.85606205214762704111567081669, 7.57796166916851333109689744262, 8.318087810712769332036019634096

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