Properties

Label 2-495-1.1-c5-0-12
Degree $2$
Conductor $495$
Sign $1$
Analytic cond. $79.3899$
Root an. cond. $8.91010$
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  − 0.694·2-s − 31.5·4-s − 25·5-s + 83.1·7-s + 44.1·8-s + 17.3·10-s − 121·11-s − 674.·13-s − 57.7·14-s + 977.·16-s − 1.92e3·17-s − 149.·19-s + 787.·20-s + 84.0·22-s + 1.35e3·23-s + 625·25-s + 468.·26-s − 2.62e3·28-s + 7.32e3·29-s − 4.21e3·31-s − 2.09e3·32-s + 1.33e3·34-s − 2.07e3·35-s − 1.34e4·37-s + 104.·38-s − 1.10e3·40-s − 2.86e3·41-s + ⋯
L(s)  = 1  − 0.122·2-s − 0.984·4-s − 0.447·5-s + 0.641·7-s + 0.243·8-s + 0.0549·10-s − 0.301·11-s − 1.10·13-s − 0.0787·14-s + 0.954·16-s − 1.61·17-s − 0.0951·19-s + 0.440·20-s + 0.0370·22-s + 0.534·23-s + 0.200·25-s + 0.135·26-s − 0.631·28-s + 1.61·29-s − 0.787·31-s − 0.361·32-s + 0.198·34-s − 0.286·35-s − 1.61·37-s + 0.0116·38-s − 0.109·40-s − 0.266·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(79.3899\)
Root analytic conductor: \(8.91010\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7959918618\)
\(L(\frac12)\) \(\approx\) \(0.7959918618\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + 25T \)
11 \( 1 + 121T \)
good2 \( 1 + 0.694T + 32T^{2} \)
7 \( 1 - 83.1T + 1.68e4T^{2} \)
13 \( 1 + 674.T + 3.71e5T^{2} \)
17 \( 1 + 1.92e3T + 1.41e6T^{2} \)
19 \( 1 + 149.T + 2.47e6T^{2} \)
23 \( 1 - 1.35e3T + 6.43e6T^{2} \)
29 \( 1 - 7.32e3T + 2.05e7T^{2} \)
31 \( 1 + 4.21e3T + 2.86e7T^{2} \)
37 \( 1 + 1.34e4T + 6.93e7T^{2} \)
41 \( 1 + 2.86e3T + 1.15e8T^{2} \)
43 \( 1 + 2.20e4T + 1.47e8T^{2} \)
47 \( 1 - 1.45e4T + 2.29e8T^{2} \)
53 \( 1 + 1.33e4T + 4.18e8T^{2} \)
59 \( 1 + 4.58e4T + 7.14e8T^{2} \)
61 \( 1 - 1.89e4T + 8.44e8T^{2} \)
67 \( 1 - 6.65e3T + 1.35e9T^{2} \)
71 \( 1 - 6.10e4T + 1.80e9T^{2} \)
73 \( 1 + 1.73e4T + 2.07e9T^{2} \)
79 \( 1 - 6.16e4T + 3.07e9T^{2} \)
83 \( 1 - 6.52e4T + 3.93e9T^{2} \)
89 \( 1 - 1.09e5T + 5.58e9T^{2} \)
97 \( 1 - 8.37e4T + 8.58e9T^{2} \)
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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

−10.16260908528484151646752069252, −9.086253849255547889045641472559, −8.469015215439844258900122145900, −7.60073584486148454846809951154, −6.59704470162356467193931530052, −4.91309294952406055387739865405, −4.78624471986705496006413912244, −3.41375113520436924607620786305, −1.98130498614868975564402576676, −0.45238787643996408916638723705, 0.45238787643996408916638723705, 1.98130498614868975564402576676, 3.41375113520436924607620786305, 4.78624471986705496006413912244, 4.91309294952406055387739865405, 6.59704470162356467193931530052, 7.60073584486148454846809951154, 8.469015215439844258900122145900, 9.086253849255547889045641472559, 10.16260908528484151646752069252

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