Properties

Label 2-495-1.1-c5-0-43
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.08·2-s − 30.8·4-s − 25·5-s − 139.·7-s + 67.9·8-s + 27.0·10-s + 121·11-s − 646.·13-s + 150.·14-s + 913.·16-s + 1.37e3·17-s + 1.90e3·19-s + 770.·20-s − 130.·22-s − 343.·23-s + 625·25-s + 698.·26-s + 4.30e3·28-s − 53.5·29-s + 634.·31-s − 3.16e3·32-s − 1.49e3·34-s + 3.48e3·35-s + 1.16e4·37-s − 2.06e3·38-s − 1.69e3·40-s − 1.88e4·41-s + ⋯
L(s)  = 1  − 0.191·2-s − 0.963·4-s − 0.447·5-s − 1.07·7-s + 0.375·8-s + 0.0854·10-s + 0.301·11-s − 1.06·13-s + 0.205·14-s + 0.891·16-s + 1.15·17-s + 1.21·19-s + 0.430·20-s − 0.0576·22-s − 0.135·23-s + 0.200·25-s + 0.202·26-s + 1.03·28-s − 0.0118·29-s + 0.118·31-s − 0.545·32-s − 0.221·34-s + 0.481·35-s + 1.40·37-s − 0.231·38-s − 0.167·40-s − 1.75·41-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: $\chi_{495} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 495,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.08T + 32T^{2} \)
7 \( 1 + 139.T + 1.68e4T^{2} \)
13 \( 1 + 646.T + 3.71e5T^{2} \)
17 \( 1 - 1.37e3T + 1.41e6T^{2} \)
19 \( 1 - 1.90e3T + 2.47e6T^{2} \)
23 \( 1 + 343.T + 6.43e6T^{2} \)
29 \( 1 + 53.5T + 2.05e7T^{2} \)
31 \( 1 - 634.T + 2.86e7T^{2} \)
37 \( 1 - 1.16e4T + 6.93e7T^{2} \)
41 \( 1 + 1.88e4T + 1.15e8T^{2} \)
43 \( 1 - 1.33e4T + 1.47e8T^{2} \)
47 \( 1 - 2.25e3T + 2.29e8T^{2} \)
53 \( 1 - 8.90e3T + 4.18e8T^{2} \)
59 \( 1 - 1.12e4T + 7.14e8T^{2} \)
61 \( 1 - 1.18e4T + 8.44e8T^{2} \)
67 \( 1 + 5.73e4T + 1.35e9T^{2} \)
71 \( 1 + 3.10e4T + 1.80e9T^{2} \)
73 \( 1 - 5.60e4T + 2.07e9T^{2} \)
79 \( 1 + 883.T + 3.07e9T^{2} \)
83 \( 1 + 9.39e4T + 3.93e9T^{2} \)
89 \( 1 + 2.17e4T + 5.58e9T^{2} \)
97 \( 1 + 1.58e5T + 8.58e9T^{2} \)
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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

−9.786933515172844498396175787468, −8.970359519836336292430568783602, −7.86838214065113783636151243362, −7.17678415380728784425510634181, −5.87178833177433471847123245910, −4.91520514685328850814074401651, −3.80870189604633262763517229755, −2.94257371112952454596774972605, −1.03121892672350698630691836067, 0, 1.03121892672350698630691836067, 2.94257371112952454596774972605, 3.80870189604633262763517229755, 4.91520514685328850814074401651, 5.87178833177433471847123245910, 7.17678415380728784425510634181, 7.86838214065113783636151243362, 8.970359519836336292430568783602, 9.786933515172844498396175787468

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