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Properties

Label 2-135-9.4-c3-0-8
Degree $2$
Conductor $135$
Sign $0.364 + 0.931i$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.785 + 1.35i)2-s + (2.76 − 4.79i)4-s + (−2.5 + 4.33i)5-s + (−17.1 − 29.6i)7-s + 21.2·8-s − 7.85·10-s + (−13.4 − 23.2i)11-s + (9.74 − 16.8i)13-s + (26.8 − 46.5i)14-s + (−5.45 − 9.44i)16-s − 29.1·17-s + 47.1·19-s + (13.8 + 23.9i)20-s + (21.1 − 36.5i)22-s + (56.6 − 98.0i)23-s + ⋯
L(s)  = 1  + (0.277 + 0.480i)2-s + (0.345 − 0.599i)4-s + (−0.223 + 0.387i)5-s + (−0.924 − 1.60i)7-s + 0.939·8-s − 0.248·10-s + (−0.368 − 0.638i)11-s + (0.207 − 0.360i)13-s + (0.513 − 0.888i)14-s + (−0.0852 − 0.147i)16-s − 0.415·17-s + 0.568·19-s + (0.154 + 0.267i)20-s + (0.204 − 0.354i)22-s + (0.513 − 0.888i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ 0.364 + 0.931i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34490 - 0.917901i\)
\(L(\frac12)\) \(\approx\) \(1.34490 - 0.917901i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T \)
good2 \( 1 + (-0.785 - 1.35i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (17.1 + 29.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (13.4 + 23.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-9.74 + 16.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 29.1T + 4.91e3T^{2} \)
19 \( 1 - 47.1T + 6.85e3T^{2} \)
23 \( 1 + (-56.6 + 98.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-40.6 - 70.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-5.41 + 9.38i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 410.T + 5.06e4T^{2} \)
41 \( 1 + (221. - 384. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (169. + 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-118. - 204. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 609.T + 1.48e5T^{2} \)
59 \( 1 + (-7.77 + 13.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (108. - 187. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 65.4T + 3.57e5T^{2} \)
73 \( 1 - 711.T + 3.89e5T^{2} \)
79 \( 1 + (-478. - 829. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (261. + 452. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + (400. + 694. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.96472352482478690303689631363, −11.19981240613554924121589787444, −10.58442948730217039481408572746, −9.723687637846696014294228351842, −7.927951651441955205341677708122, −6.93207646878636794963938403296, −6.18352210258732627728887051043, −4.61290478085282083459939771528, −3.17211634216235753804944431489, −0.75165554421958782543530015710, 2.18094464923114990922160779186, 3.35984341923880730459622291873, 4.92463889962973102707938560520, 6.33252119734167546233590749731, 7.65541762607207868878530056134, 8.856451114483060320543867174799, 9.773590350475349677594685765022, 11.30801928991678199402164865678, 12.04293609663073082103025665955, 12.75002083276482476709717257169

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