Properties

Label 2-1008-1.1-c5-0-14
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  − 50.0·5-s + 49·7-s − 150.·11-s + 522.·13-s + 345.·17-s + 2.14e3·19-s + 484.·23-s − 618.·25-s − 3.94e3·29-s + 3.90e3·31-s − 2.45e3·35-s − 5.57e3·37-s − 7.70e3·41-s − 2.56e3·43-s − 2.09e4·47-s + 2.40e3·49-s + 2.96e4·53-s + 7.53e3·55-s − 1.48e4·59-s + 4.98e3·61-s − 2.61e4·65-s − 3.59e4·67-s − 1.31e4·71-s + 4.91e4·73-s − 7.37e3·77-s + 1.72e4·79-s + 1.79e4·83-s + ⋯
L(s)  = 1  − 0.895·5-s + 0.377·7-s − 0.374·11-s + 0.857·13-s + 0.289·17-s + 1.36·19-s + 0.191·23-s − 0.197·25-s − 0.871·29-s + 0.729·31-s − 0.338·35-s − 0.669·37-s − 0.715·41-s − 0.211·43-s − 1.38·47-s + 0.142·49-s + 1.45·53-s + 0.335·55-s − 0.556·59-s + 0.171·61-s − 0.767·65-s − 0.977·67-s − 0.310·71-s + 1.07·73-s − 0.141·77-s + 0.311·79-s + 0.285·83-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

\(L(3)\) \(\approx\) \(1.807409967\)
\(L(\frac12)\) \(\approx\) \(1.807409967\)
\(L(\frac{7}{2})\) 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 \( 1 + 50.0T + 3.12e3T^{2} \)
11 \( 1 + 150.T + 1.61e5T^{2} \)
13 \( 1 - 522.T + 3.71e5T^{2} \)
17 \( 1 - 345.T + 1.41e6T^{2} \)
19 \( 1 - 2.14e3T + 2.47e6T^{2} \)
23 \( 1 - 484.T + 6.43e6T^{2} \)
29 \( 1 + 3.94e3T + 2.05e7T^{2} \)
31 \( 1 - 3.90e3T + 2.86e7T^{2} \)
37 \( 1 + 5.57e3T + 6.93e7T^{2} \)
41 \( 1 + 7.70e3T + 1.15e8T^{2} \)
43 \( 1 + 2.56e3T + 1.47e8T^{2} \)
47 \( 1 + 2.09e4T + 2.29e8T^{2} \)
53 \( 1 - 2.96e4T + 4.18e8T^{2} \)
59 \( 1 + 1.48e4T + 7.14e8T^{2} \)
61 \( 1 - 4.98e3T + 8.44e8T^{2} \)
67 \( 1 + 3.59e4T + 1.35e9T^{2} \)
71 \( 1 + 1.31e4T + 1.80e9T^{2} \)
73 \( 1 - 4.91e4T + 2.07e9T^{2} \)
79 \( 1 - 1.72e4T + 3.07e9T^{2} \)
83 \( 1 - 1.79e4T + 3.93e9T^{2} \)
89 \( 1 - 7.14e4T + 5.58e9T^{2} \)
97 \( 1 - 7.47e4T + 8.58e9T^{2} \)
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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

−9.153394181357466592388688580762, −8.229131441260520406316309998579, −7.71207520409230729006815982749, −6.84652955595899099619094034149, −5.70489701448378980555468616391, −4.90040981801038448679998021240, −3.81822274462650904592590653002, −3.12007537402998256553873533521, −1.68178219230447011041899347458, −0.60129495428075790895057899327, 0.60129495428075790895057899327, 1.68178219230447011041899347458, 3.12007537402998256553873533521, 3.81822274462650904592590653002, 4.90040981801038448679998021240, 5.70489701448378980555468616391, 6.84652955595899099619094034149, 7.71207520409230729006815982749, 8.229131441260520406316309998579, 9.153394181357466592388688580762

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