Properties

Label 2-1008-1.1-c5-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 70.4·5-s − 49·7-s − 169.·11-s − 10.8·13-s − 2.32e3·17-s + 1.50e3·19-s + 3.75e3·23-s + 1.83e3·25-s − 5.64e3·29-s − 4.28e3·31-s − 3.45e3·35-s + 36.5·37-s − 90.6·41-s + 8.29e3·43-s + 1.85e4·47-s + 2.40e3·49-s + 1.18e4·53-s − 1.19e4·55-s + 4.82e4·59-s + 2.22e4·61-s − 764.·65-s + 1.00e4·67-s + 4.17e4·71-s + 6.71e4·73-s + 8.29e3·77-s − 5.83e4·79-s − 9.94e4·83-s + ⋯
L(s)  = 1  + 1.25·5-s − 0.377·7-s − 0.421·11-s − 0.0178·13-s − 1.95·17-s + 0.958·19-s + 1.48·23-s + 0.587·25-s − 1.24·29-s − 0.800·31-s − 0.476·35-s + 0.00438·37-s − 0.00842·41-s + 0.683·43-s + 1.22·47-s + 0.142·49-s + 0.580·53-s − 0.531·55-s + 1.80·59-s + 0.764·61-s − 0.0224·65-s + 0.274·67-s + 0.984·71-s + 1.47·73-s + 0.159·77-s − 1.05·79-s − 1.58·83-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: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.507253394\)
\(L(\frac12)\) \(\approx\) \(2.507253394\)
\(L(\frac{7}{2})\) 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 \( 1 - 70.4T + 3.12e3T^{2} \)
11 \( 1 + 169.T + 1.61e5T^{2} \)
13 \( 1 + 10.8T + 3.71e5T^{2} \)
17 \( 1 + 2.32e3T + 1.41e6T^{2} \)
19 \( 1 - 1.50e3T + 2.47e6T^{2} \)
23 \( 1 - 3.75e3T + 6.43e6T^{2} \)
29 \( 1 + 5.64e3T + 2.05e7T^{2} \)
31 \( 1 + 4.28e3T + 2.86e7T^{2} \)
37 \( 1 - 36.5T + 6.93e7T^{2} \)
41 \( 1 + 90.6T + 1.15e8T^{2} \)
43 \( 1 - 8.29e3T + 1.47e8T^{2} \)
47 \( 1 - 1.85e4T + 2.29e8T^{2} \)
53 \( 1 - 1.18e4T + 4.18e8T^{2} \)
59 \( 1 - 4.82e4T + 7.14e8T^{2} \)
61 \( 1 - 2.22e4T + 8.44e8T^{2} \)
67 \( 1 - 1.00e4T + 1.35e9T^{2} \)
71 \( 1 - 4.17e4T + 1.80e9T^{2} \)
73 \( 1 - 6.71e4T + 2.07e9T^{2} \)
79 \( 1 + 5.83e4T + 3.07e9T^{2} \)
83 \( 1 + 9.94e4T + 3.93e9T^{2} \)
89 \( 1 - 1.07e5T + 5.58e9T^{2} \)
97 \( 1 + 9.64e4T + 8.58e9T^{2} \)
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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

−9.254587649706692344170659737037, −8.708529799704381972108460860644, −7.32732889653923515081195896568, −6.73664207031382757215138944125, −5.70956035145050303436706763992, −5.14446048769584502440866744108, −3.90800501490904085827915141929, −2.66281851428337519919138822866, −1.96817828290049143746193664862, −0.67483672312843545708990423496, 0.67483672312843545708990423496, 1.96817828290049143746193664862, 2.66281851428337519919138822866, 3.90800501490904085827915141929, 5.14446048769584502440866744108, 5.70956035145050303436706763992, 6.73664207031382757215138944125, 7.32732889653923515081195896568, 8.708529799704381972108460860644, 9.254587649706692344170659737037

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