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Properties

Label 2-135-9.7-c3-0-7
Degree $2$
Conductor $135$
Sign $0.790 + 0.612i$
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  + (−2.13 + 3.69i)2-s + (−5.08 − 8.80i)4-s + (−2.5 − 4.33i)5-s + (−15.3 + 26.6i)7-s + 9.21·8-s + 21.3·10-s + (20.3 − 35.2i)11-s + (−31.6 − 54.7i)13-s + (−65.5 − 113. i)14-s + (21.0 − 36.3i)16-s + 6.58·17-s + 75.3·19-s + (−25.4 + 44.0i)20-s + (86.7 + 150. i)22-s + (−31.1 − 54.0i)23-s + ⋯
L(s)  = 1  + (−0.753 + 1.30i)2-s + (−0.635 − 1.10i)4-s + (−0.223 − 0.387i)5-s + (−0.830 + 1.43i)7-s + 0.407·8-s + 0.673·10-s + (0.557 − 0.966i)11-s + (−0.674 − 1.16i)13-s + (−1.25 − 2.16i)14-s + (0.328 − 0.568i)16-s + 0.0940·17-s + 0.910·19-s + (−0.284 + 0.492i)20-s + (0.840 + 1.45i)22-s + (−0.282 − 0.489i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.370857 - 0.126932i\)
\(L(\frac12)\) \(\approx\) \(0.370857 - 0.126932i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.13 - 3.69i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (15.3 - 26.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-20.3 + 35.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (31.6 + 54.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 6.58T + 4.91e3T^{2} \)
19 \( 1 - 75.3T + 6.85e3T^{2} \)
23 \( 1 + (31.1 + 54.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-24.8 + 42.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (51.5 + 89.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 282.T + 5.06e4T^{2} \)
41 \( 1 + (78.7 + 136. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-168. + 292. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-22.2 + 38.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 26.2T + 1.48e5T^{2} \)
59 \( 1 + (212. + 368. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (425. - 736. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (48.1 + 83.4i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 952.T + 3.57e5T^{2} \)
73 \( 1 + 50.8T + 3.89e5T^{2} \)
79 \( 1 + (-98.6 + 170. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (98.8 - 171. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (-715. + 1.23e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.55045954120939524218446373403, −11.84932565697942635348300707530, −10.09580111225124839399954499898, −9.078770507130756231654070396627, −8.498110473784730300658347897386, −7.34938051293053760996108788248, −6.02299161073818994884114103905, −5.42386826011015921601053329419, −3.08126476195134805254451166958, −0.26070276594236223252332383942, 1.49146297106141624297562210896, 3.21986589507684364907138090050, 4.29812270362265075487223999736, 6.70500903582261906142711151125, 7.53187824258648057840038965378, 9.277344947679938364062164049907, 9.849151831391239316782106134205, 10.67828318334030678221183981891, 11.72894822724864165969341271918, 12.46010454848945249839785636970

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