Properties

Label 2-495-1.1-c3-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.03·2-s + 17.3·4-s + 5·5-s + 18.5·7-s − 46.8·8-s − 25.1·10-s + 11·11-s + 47.0·13-s − 93.4·14-s + 97.2·16-s − 2.62·17-s + 157.·19-s + 86.5·20-s − 55.3·22-s + 186.·23-s + 25·25-s − 236.·26-s + 321.·28-s − 270.·29-s − 25.7·31-s − 114.·32-s + 13.2·34-s + 92.8·35-s + 228.·37-s − 791.·38-s − 234.·40-s + 6.85·41-s + ⋯
L(s)  = 1  − 1.77·2-s + 2.16·4-s + 0.447·5-s + 1.00·7-s − 2.07·8-s − 0.795·10-s + 0.301·11-s + 1.00·13-s − 1.78·14-s + 1.51·16-s − 0.0374·17-s + 1.89·19-s + 0.967·20-s − 0.536·22-s + 1.69·23-s + 0.200·25-s − 1.78·26-s + 2.17·28-s − 1.73·29-s − 0.149·31-s − 0.632·32-s + 0.0666·34-s + 0.448·35-s + 1.01·37-s − 3.37·38-s − 0.926·40-s + 0.0261·41-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: \(0\)
Selberg data: \((2,\ 495,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.212355279\)
\(L(\frac12)\) \(\approx\) \(1.212355279\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.03T + 8T^{2} \)
7 \( 1 - 18.5T + 343T^{2} \)
13 \( 1 - 47.0T + 2.19e3T^{2} \)
17 \( 1 + 2.62T + 4.91e3T^{2} \)
19 \( 1 - 157.T + 6.85e3T^{2} \)
23 \( 1 - 186.T + 1.21e4T^{2} \)
29 \( 1 + 270.T + 2.43e4T^{2} \)
31 \( 1 + 25.7T + 2.97e4T^{2} \)
37 \( 1 - 228.T + 5.06e4T^{2} \)
41 \( 1 - 6.85T + 6.89e4T^{2} \)
43 \( 1 + 386.T + 7.95e4T^{2} \)
47 \( 1 - 283.T + 1.03e5T^{2} \)
53 \( 1 + 452.T + 1.48e5T^{2} \)
59 \( 1 + 88.1T + 2.05e5T^{2} \)
61 \( 1 + 829.T + 2.26e5T^{2} \)
67 \( 1 - 489.T + 3.00e5T^{2} \)
71 \( 1 + 63.5T + 3.57e5T^{2} \)
73 \( 1 + 78.3T + 3.89e5T^{2} \)
79 \( 1 + 203.T + 4.93e5T^{2} \)
83 \( 1 - 287.T + 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 62.6T + 9.12e5T^{2} \)
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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

−10.42707914841529397759243528271, −9.340714796609657117080836721231, −9.007621329551848111539003218984, −7.901485130070990371599838669848, −7.32106848077791155774473429886, −6.20269390718508695468669645326, −5.07607615636543250427674090518, −3.22165704453983294685945421739, −1.73166626922966463216450965558, −0.987310140794372593341620285260, 0.987310140794372593341620285260, 1.73166626922966463216450965558, 3.22165704453983294685945421739, 5.07607615636543250427674090518, 6.20269390718508695468669645326, 7.32106848077791155774473429886, 7.901485130070990371599838669848, 9.007621329551848111539003218984, 9.340714796609657117080836721231, 10.42707914841529397759243528271

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