Properties

Label 2-135-135.113-c3-0-15
Degree $2$
Conductor $135$
Sign $0.942 - 0.333i$
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  + (1.17 + 0.102i)2-s + (−5.05 + 1.20i)3-s + (−6.51 − 1.14i)4-s + (−6.83 − 8.84i)5-s + (−6.05 + 0.894i)6-s + (27.4 + 19.2i)7-s + (−16.6 − 4.45i)8-s + (24.0 − 12.1i)9-s + (−7.11 − 11.0i)10-s + (1.45 − 4.00i)11-s + (34.3 − 2.03i)12-s + (3.68 + 42.1i)13-s + (30.2 + 25.3i)14-s + (45.2 + 36.4i)15-s + (30.6 + 11.1i)16-s + (72.5 − 19.4i)17-s + ⋯
L(s)  = 1  + (0.414 + 0.0362i)2-s + (−0.972 + 0.231i)3-s + (−0.814 − 0.143i)4-s + (−0.611 − 0.791i)5-s + (−0.411 + 0.0608i)6-s + (1.48 + 1.03i)7-s + (−0.734 − 0.196i)8-s + (0.892 − 0.450i)9-s + (−0.224 − 0.350i)10-s + (0.0399 − 0.109i)11-s + (0.825 − 0.0490i)12-s + (0.0786 + 0.899i)13-s + (0.577 + 0.484i)14-s + (0.778 + 0.628i)15-s + (0.479 + 0.174i)16-s + (1.03 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.22383 + 0.210344i\)
\(L(\frac12)\) \(\approx\) \(1.22383 + 0.210344i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.05 - 1.20i)T \)
5 \( 1 + (6.83 + 8.84i)T \)
good2 \( 1 + (-1.17 - 0.102i)T + (7.87 + 1.38i)T^{2} \)
7 \( 1 + (-27.4 - 19.2i)T + (117. + 322. i)T^{2} \)
11 \( 1 + (-1.45 + 4.00i)T + (-1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-3.68 - 42.1i)T + (-2.16e3 + 381. i)T^{2} \)
17 \( 1 + (-72.5 + 19.4i)T + (4.25e3 - 2.45e3i)T^{2} \)
19 \( 1 + (-115. + 66.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-33.5 - 47.8i)T + (-4.16e3 + 1.14e4i)T^{2} \)
29 \( 1 + (127. - 107. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-1.16 + 6.61i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (28.8 + 107. i)T + (-4.38e4 + 2.53e4i)T^{2} \)
41 \( 1 + (-130. + 155. i)T + (-1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 + (-73.9 - 158. i)T + (-5.11e4 + 6.09e4i)T^{2} \)
47 \( 1 + (-270. + 386. i)T + (-3.55e4 - 9.75e4i)T^{2} \)
53 \( 1 + (-311. + 311. i)T - 1.48e5iT^{2} \)
59 \( 1 + (550. - 200. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-72.8 - 413. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-869. + 76.0i)T + (2.96e5 - 5.22e4i)T^{2} \)
71 \( 1 + (-421. - 243. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (190. - 712. i)T + (-3.36e5 - 1.94e5i)T^{2} \)
79 \( 1 + (200. + 238. i)T + (-8.56e4 + 4.85e5i)T^{2} \)
83 \( 1 + (14.5 - 166. i)T + (-5.63e5 - 9.92e4i)T^{2} \)
89 \( 1 + (-663. - 1.14e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (830. - 387. i)T + (5.86e5 - 6.99e5i)T^{2} \)
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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

−12.52731090888862338463007923903, −11.83414948260092432682116045208, −11.24160389270216779359162063940, −9.486733808675939452561050589245, −8.787355668494980312210595955372, −7.46225920684188418689675779661, −5.36685054887526034349760967230, −5.23788911582526830899069284389, −3.98496625083622596194915155344, −1.09802990970212867127389860954, 0.901460023175320612340434449990, 3.64529291243525642780793122744, 4.73991295996122469300397318705, 5.79265567788034714234048057323, 7.56905840219526368078959800612, 7.903056653225910973548244258944, 9.988394022394225456008739568052, 10.83724851014002898685221971718, 11.71612122093614711753255416138, 12.56278403319200475914484615205

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