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Properties

Label 2-135-9.2-c2-0-6
Degree $2$
Conductor $135$
Sign $0.857 + 0.515i$
Analytic cond. $3.67848$
Root an. cond. $1.91793$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.20 + 0.696i)2-s + (−1.03 − 1.78i)4-s + (1.93 − 1.11i)5-s + (4.41 − 7.63i)7-s − 8.43i·8-s + 3.11·10-s + (0.805 + 0.464i)11-s + (12.2 + 21.2i)13-s + (10.6 − 6.13i)14-s + (1.74 − 3.02i)16-s − 18.3i·17-s − 5.58·19-s + (−3.99 − 2.30i)20-s + (0.647 + 1.12i)22-s + (−20.6 + 11.9i)23-s + ⋯
L(s)  = 1  + (0.602 + 0.348i)2-s + (−0.257 − 0.446i)4-s + (0.387 − 0.223i)5-s + (0.630 − 1.09i)7-s − 1.05i·8-s + 0.311·10-s + (0.0732 + 0.0422i)11-s + (0.941 + 1.63i)13-s + (0.759 − 0.438i)14-s + (0.109 − 0.189i)16-s − 1.07i·17-s − 0.294·19-s + (−0.199 − 0.115i)20-s + (0.0294 + 0.0509i)22-s + (−0.896 + 0.517i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.857 + 0.515i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ 0.857 + 0.515i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88258 - 0.522108i\)
\(L(\frac12)\) \(\approx\) \(1.88258 - 0.522108i\)
\(L(2)\) 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 \( 1 + (-1.93 + 1.11i)T \)
good2 \( 1 + (-1.20 - 0.696i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (-4.41 + 7.63i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.805 - 0.464i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-12.2 - 21.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 18.3iT - 289T^{2} \)
19 \( 1 + 5.58T + 361T^{2} \)
23 \( 1 + (20.6 - 11.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-23.7 - 13.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4.66 - 8.08i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 24.7T + 1.36e3T^{2} \)
41 \( 1 + (-6.45 + 3.72i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (17.7 - 30.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-0.298 - 0.172i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 81.8iT - 2.80e3T^{2} \)
59 \( 1 + (-65.9 + 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (29.6 - 51.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-40.9 - 70.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 37.5iT - 5.04e3T^{2} \)
73 \( 1 - 3.49T + 5.32e3T^{2} \)
79 \( 1 + (-62.0 + 107. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (48.4 + 27.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 6.78iT - 7.92e3T^{2} \)
97 \( 1 + (58.5 - 101. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.42066315470652729938550026438, −11.95600603510910845822670465172, −10.85134178806060629329550409445, −9.806232000759436210717601973781, −8.784188858945072019849629536038, −7.20551085148302944873469316057, −6.25446288115126797593064103616, −4.86125634628406868162116686556, −4.00926817691077704140558602896, −1.35471549355657075435874843175, 2.28019808284395159287571531070, 3.65886692876207728859480742954, 5.20333176314741768373667400159, 6.10858991554251804480462767152, 8.182813803784281634144464662560, 8.523398671238326580458350371814, 10.21265878587372483179982150524, 11.22721390049411377638978958685, 12.26944829243329541739041622504, 12.95018545453562581810998345820

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