Properties

Label 2-135-1.1-c3-0-0
Degree $2$
Conductor $135$
Sign $1$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
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  − 4.45·2-s + 11.8·4-s − 5·5-s + 5.08·7-s − 17.3·8-s + 22.2·10-s − 58.3·11-s + 21.2·13-s − 22.6·14-s − 17.8·16-s + 68.8·17-s − 40.8·19-s − 59.4·20-s + 259.·22-s + 144.·23-s + 25·25-s − 94.5·26-s + 60.3·28-s + 220.·29-s + 291.·31-s + 218.·32-s − 307.·34-s − 25.4·35-s + 260.·37-s + 182.·38-s + 86.6·40-s + 169.·41-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.48·4-s − 0.447·5-s + 0.274·7-s − 0.765·8-s + 0.705·10-s − 1.59·11-s + 0.452·13-s − 0.432·14-s − 0.278·16-s + 0.982·17-s − 0.492·19-s − 0.664·20-s + 2.51·22-s + 1.30·23-s + 0.200·25-s − 0.713·26-s + 0.407·28-s + 1.40·29-s + 1.68·31-s + 1.20·32-s − 1.54·34-s − 0.122·35-s + 1.15·37-s + 0.776·38-s + 0.342·40-s + 0.646·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6425282407\)
\(L(\frac12)\) \(\approx\) \(0.6425282407\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
good2 \( 1 + 4.45T + 8T^{2} \)
7 \( 1 - 5.08T + 343T^{2} \)
11 \( 1 + 58.3T + 1.33e3T^{2} \)
13 \( 1 - 21.2T + 2.19e3T^{2} \)
17 \( 1 - 68.8T + 4.91e3T^{2} \)
19 \( 1 + 40.8T + 6.85e3T^{2} \)
23 \( 1 - 144.T + 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 - 291.T + 2.97e4T^{2} \)
37 \( 1 - 260.T + 5.06e4T^{2} \)
41 \( 1 - 169.T + 6.89e4T^{2} \)
43 \( 1 + 438.T + 7.95e4T^{2} \)
47 \( 1 - 255.T + 1.03e5T^{2} \)
53 \( 1 + 214.T + 1.48e5T^{2} \)
59 \( 1 + 331.T + 2.05e5T^{2} \)
61 \( 1 - 54.9T + 2.26e5T^{2} \)
67 \( 1 - 758.T + 3.00e5T^{2} \)
71 \( 1 - 904.T + 3.57e5T^{2} \)
73 \( 1 - 866.T + 3.89e5T^{2} \)
79 \( 1 - 206.T + 4.93e5T^{2} \)
83 \( 1 - 463.T + 5.71e5T^{2} \)
89 \( 1 + 601.T + 7.04e5T^{2} \)
97 \( 1 - 229.T + 9.12e5T^{2} \)
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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

−12.53097571097461536359890796374, −11.26970250011170405448771694661, −10.55181642592999060164628626277, −9.665427418684236906183210959972, −8.253434029612455221632646646446, −7.975669044862759349257491451064, −6.63608848599958390522406247067, −4.90389355221207762736139942573, −2.75516325646685517928478298989, −0.853715729164523564642244320077, 0.853715729164523564642244320077, 2.75516325646685517928478298989, 4.90389355221207762736139942573, 6.63608848599958390522406247067, 7.975669044862759349257491451064, 8.253434029612455221632646646446, 9.665427418684236906183210959972, 10.55181642592999060164628626277, 11.26970250011170405448771694661, 12.53097571097461536359890796374

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