Properties

Label 2-495-1.1-c3-0-33
Degree $2$
Conductor $495$
Sign $-1$
Analytic cond. $29.2059$
Root an. cond. $5.40425$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s − 5.56·4-s + 5·5-s − 10.2·7-s + 21.1·8-s − 7.80·10-s + 11·11-s − 40.8·13-s + 16·14-s + 11.4·16-s + 98.7·17-s − 39.6·19-s − 27.8·20-s − 17.1·22-s − 61.6·23-s + 25·25-s + 63.8·26-s + 56.9·28-s + 149.·29-s + 54.7·31-s − 187.·32-s − 154.·34-s − 51.2·35-s + 44.8·37-s + 61.9·38-s + 105.·40-s − 336.·41-s + ⋯
L(s)  = 1  − 0.552·2-s − 0.695·4-s + 0.447·5-s − 0.553·7-s + 0.935·8-s − 0.246·10-s + 0.301·11-s − 0.872·13-s + 0.305·14-s + 0.178·16-s + 1.40·17-s − 0.478·19-s − 0.310·20-s − 0.166·22-s − 0.559·23-s + 0.200·25-s + 0.481·26-s + 0.384·28-s + 0.954·29-s + 0.317·31-s − 1.03·32-s − 0.777·34-s − 0.247·35-s + 0.199·37-s + 0.264·38-s + 0.418·40-s − 1.28·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(29.2059\)
Root analytic conductor: \(5.40425\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 495,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 5T \)
11 \( 1 - 11T \)
good2 \( 1 + 1.56T + 8T^{2} \)
7 \( 1 + 10.2T + 343T^{2} \)
13 \( 1 + 40.8T + 2.19e3T^{2} \)
17 \( 1 - 98.7T + 4.91e3T^{2} \)
19 \( 1 + 39.6T + 6.85e3T^{2} \)
23 \( 1 + 61.6T + 1.21e4T^{2} \)
29 \( 1 - 149.T + 2.43e4T^{2} \)
31 \( 1 - 54.7T + 2.97e4T^{2} \)
37 \( 1 - 44.8T + 5.06e4T^{2} \)
41 \( 1 + 336.T + 6.89e4T^{2} \)
43 \( 1 + 2.36T + 7.95e4T^{2} \)
47 \( 1 - 333.T + 1.03e5T^{2} \)
53 \( 1 + 640.T + 1.48e5T^{2} \)
59 \( 1 - 370.T + 2.05e5T^{2} \)
61 \( 1 + 714.T + 2.26e5T^{2} \)
67 \( 1 + 404.T + 3.00e5T^{2} \)
71 \( 1 + 939.T + 3.57e5T^{2} \)
73 \( 1 + 362.T + 3.89e5T^{2} \)
79 \( 1 - 951.T + 4.93e5T^{2} \)
83 \( 1 + 735.T + 5.71e5T^{2} \)
89 \( 1 + 385.T + 7.04e5T^{2} \)
97 \( 1 + 966.T + 9.12e5T^{2} \)
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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

−9.948163573622847215252595845269, −9.371755726495176599387817081107, −8.383063868392147302400508649267, −7.54440657040370455372181697887, −6.41797549725975424547959293615, −5.33945680527724516925475748585, −4.31975899509896448404227576913, −3.01925994101984347803007631576, −1.41939565351603278287198016777, 0, 1.41939565351603278287198016777, 3.01925994101984347803007631576, 4.31975899509896448404227576913, 5.33945680527724516925475748585, 6.41797549725975424547959293615, 7.54440657040370455372181697887, 8.383063868392147302400508649267, 9.371755726495176599387817081107, 9.948163573622847215252595845269

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