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Properties

Label 2-135-5.4-c3-0-22
Degree $2$
Conductor $135$
Sign $-0.735 - 0.677i$
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  − 4.92i·2-s − 16.2·4-s + (7.57 − 8.22i)5-s − 34.7i·7-s + 40.7i·8-s + (−40.5 − 37.3i)10-s + 12.4·11-s + 37.1i·13-s − 171.·14-s + 70.7·16-s + 102. i·17-s − 63.7·19-s + (−123. + 133. i)20-s − 61.3i·22-s − 100. i·23-s + ⋯
L(s)  = 1  − 1.74i·2-s − 2.03·4-s + (0.677 − 0.735i)5-s − 1.87i·7-s + 1.80i·8-s + (−1.28 − 1.17i)10-s + 0.341·11-s + 0.791i·13-s − 3.27·14-s + 1.10·16-s + 1.46i·17-s − 0.770·19-s + (−1.37 + 1.49i)20-s − 0.594i·22-s − 0.911i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ -0.735 - 0.677i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.516187 + 1.32295i\)
\(L(\frac12)\) \(\approx\) \(0.516187 + 1.32295i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.57 + 8.22i)T \)
good2 \( 1 + 4.92iT - 8T^{2} \)
7 \( 1 + 34.7iT - 343T^{2} \)
11 \( 1 - 12.4T + 1.33e3T^{2} \)
13 \( 1 - 37.1iT - 2.19e3T^{2} \)
17 \( 1 - 102. iT - 4.91e3T^{2} \)
19 \( 1 + 63.7T + 6.85e3T^{2} \)
23 \( 1 + 100. iT - 1.21e4T^{2} \)
29 \( 1 - 138.T + 2.43e4T^{2} \)
31 \( 1 - 24.8T + 2.97e4T^{2} \)
37 \( 1 + 186. iT - 5.06e4T^{2} \)
41 \( 1 - 255.T + 6.89e4T^{2} \)
43 \( 1 + 108. iT - 7.95e4T^{2} \)
47 \( 1 + 223. iT - 1.03e5T^{2} \)
53 \( 1 - 40.5iT - 1.48e5T^{2} \)
59 \( 1 - 131.T + 2.05e5T^{2} \)
61 \( 1 + 336.T + 2.26e5T^{2} \)
67 \( 1 + 17.8iT - 3.00e5T^{2} \)
71 \( 1 - 992.T + 3.57e5T^{2} \)
73 \( 1 + 769. iT - 3.89e5T^{2} \)
79 \( 1 - 252.T + 4.93e5T^{2} \)
83 \( 1 + 234. iT - 5.71e5T^{2} \)
89 \( 1 - 405.T + 7.04e5T^{2} \)
97 \( 1 - 1.06e3iT - 9.12e5T^{2} \)
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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

−12.21339010986951573575042151821, −10.80731180487386475734844922787, −10.41243657644553844010801899262, −9.390110503210672467216963088466, −8.325933528662610108952374377911, −6.53043864715225518025052084056, −4.52886905160573631267956404368, −3.90567627275610069496618238186, −1.91001436251146243843278528126, −0.75014705199060930501036316851, 2.73377137483200058383581032238, 5.06286500067646609913589061440, 5.86913598682891371169612053241, 6.68071578037463100161981414389, 7.952158196555143372145010798101, 9.011902833071406775775036680901, 9.724578264496487369213669460274, 11.43710358183722335460277206523, 12.70488448357950231436858803276, 13.81596146302910677206741880345

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