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Properties

Label 2-135-5.4-c3-0-11
Degree $2$
Conductor $135$
Sign $0.587 - 0.808i$
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  + 1.25i·2-s + 6.43·4-s + (9.04 + 6.57i)5-s − 5.67i·7-s + 18.0i·8-s + (−8.22 + 11.3i)10-s + 21.2·11-s − 40.9i·13-s + 7.09·14-s + 28.8·16-s + 59.1i·17-s − 21.8·19-s + (58.1 + 42.2i)20-s + 26.5i·22-s + 98.7i·23-s + ⋯
L(s)  = 1  + 0.442i·2-s + 0.804·4-s + (0.808 + 0.587i)5-s − 0.306i·7-s + 0.798i·8-s + (−0.260 + 0.357i)10-s + 0.582·11-s − 0.873i·13-s + 0.135·14-s + 0.451·16-s + 0.844i·17-s − 0.264·19-s + (0.650 + 0.472i)20-s + 0.257i·22-s + 0.895i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.587 - 0.808i$
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: \((2,\ 135,\ (\ :3/2),\ 0.587 - 0.808i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.03979 + 1.03928i\)
\(L(\frac12)\) \(\approx\) \(2.03979 + 1.03928i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.04 - 6.57i)T \)
good2 \( 1 - 1.25iT - 8T^{2} \)
7 \( 1 + 5.67iT - 343T^{2} \)
11 \( 1 - 21.2T + 1.33e3T^{2} \)
13 \( 1 + 40.9iT - 2.19e3T^{2} \)
17 \( 1 - 59.1iT - 4.91e3T^{2} \)
19 \( 1 + 21.8T + 6.85e3T^{2} \)
23 \( 1 - 98.7iT - 1.21e4T^{2} \)
29 \( 1 - 159.T + 2.43e4T^{2} \)
31 \( 1 + 69.4T + 2.97e4T^{2} \)
37 \( 1 + 235. iT - 5.06e4T^{2} \)
41 \( 1 + 491.T + 6.89e4T^{2} \)
43 \( 1 - 95.3iT - 7.95e4T^{2} \)
47 \( 1 + 548. iT - 1.03e5T^{2} \)
53 \( 1 + 509. iT - 1.48e5T^{2} \)
59 \( 1 + 741.T + 2.05e5T^{2} \)
61 \( 1 - 387.T + 2.26e5T^{2} \)
67 \( 1 + 1.06e3iT - 3.00e5T^{2} \)
71 \( 1 + 508.T + 3.57e5T^{2} \)
73 \( 1 - 914. iT - 3.89e5T^{2} \)
79 \( 1 + 925.T + 4.93e5T^{2} \)
83 \( 1 + 708. iT - 5.71e5T^{2} \)
89 \( 1 + 273.T + 7.04e5T^{2} \)
97 \( 1 + 1.45e3iT - 9.12e5T^{2} \)
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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

−13.02863176138620671915555767849, −11.79926988186204749757101077745, −10.72983127101574130237423565352, −10.04368157019075043477519641513, −8.538345567555276159657827724092, −7.30019002891643407215685485827, −6.40382023882431636665619000317, −5.44769605569783969264646955152, −3.35661495010727572484129985376, −1.81293435142188697449977708467, 1.39165960109928252916440926365, 2.70155314389181248524751849320, 4.55472286711978871102625153834, 6.07629066091307032474423732404, 6.95916264491717869071968518537, 8.624079386795526651707932797955, 9.578647979657938110862400662841, 10.55763492629402437646495663780, 11.77695357417145039857258649354, 12.33319875988286135581835263058

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