Properties

Label 2-507-1.1-c3-0-12
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.36·2-s − 3·3-s − 2.42·4-s − 6.42·5-s + 7.08·6-s + 29.4·7-s + 24.6·8-s + 9·9-s + 15.1·10-s + 0.624·11-s + 7.26·12-s − 69.6·14-s + 19.2·15-s − 38.7·16-s + 87.7·17-s − 21.2·18-s − 82.8·19-s + 15.5·20-s − 88.4·21-s − 1.47·22-s − 74.7·23-s − 73.8·24-s − 83.7·25-s − 27·27-s − 71.4·28-s + 226.·29-s − 45.5·30-s + ⋯
L(s)  = 1  − 0.835·2-s − 0.577·3-s − 0.302·4-s − 0.574·5-s + 0.482·6-s + 1.59·7-s + 1.08·8-s + 0.333·9-s + 0.479·10-s + 0.0171·11-s + 0.174·12-s − 1.32·14-s + 0.331·15-s − 0.605·16-s + 1.25·17-s − 0.278·18-s − 0.999·19-s + 0.173·20-s − 0.919·21-s − 0.0142·22-s − 0.678·23-s − 0.628·24-s − 0.670·25-s − 0.192·27-s − 0.482·28-s + 1.44·29-s − 0.276·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8769066822\)
\(L(\frac12)\) \(\approx\) \(0.8769066822\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 + 2.36T + 8T^{2} \)
5 \( 1 + 6.42T + 125T^{2} \)
7 \( 1 - 29.4T + 343T^{2} \)
11 \( 1 - 0.624T + 1.33e3T^{2} \)
17 \( 1 - 87.7T + 4.91e3T^{2} \)
19 \( 1 + 82.8T + 6.85e3T^{2} \)
23 \( 1 + 74.7T + 1.21e4T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 + 173.T + 2.97e4T^{2} \)
37 \( 1 + 112.T + 5.06e4T^{2} \)
41 \( 1 - 267.T + 6.89e4T^{2} \)
43 \( 1 - 383.T + 7.95e4T^{2} \)
47 \( 1 + 337.T + 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 - 529.T + 2.05e5T^{2} \)
61 \( 1 - 203.T + 2.26e5T^{2} \)
67 \( 1 + 121.T + 3.00e5T^{2} \)
71 \( 1 - 661.T + 3.57e5T^{2} \)
73 \( 1 + 167.T + 3.89e5T^{2} \)
79 \( 1 + 101.T + 4.93e5T^{2} \)
83 \( 1 - 506.T + 5.71e5T^{2} \)
89 \( 1 + 1.40e3T + 7.04e5T^{2} \)
97 \( 1 + 1.90e3T + 9.12e5T^{2} \)
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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

−10.54475776489834784152005792517, −9.676039661550176373003845255371, −8.463491136161441194477471211524, −8.016913113431505876496606466256, −7.23929082655130372529532620872, −5.73363883970747809119022688912, −4.75199752650937116536138547260, −3.98940217673127244186006041620, −1.84128137845546088408158905928, −0.71098487746157144149232049275, 0.71098487746157144149232049275, 1.84128137845546088408158905928, 3.98940217673127244186006041620, 4.75199752650937116536138547260, 5.73363883970747809119022688912, 7.23929082655130372529532620872, 8.016913113431505876496606466256, 8.463491136161441194477471211524, 9.676039661550176373003845255371, 10.54475776489834784152005792517

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