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Properties

Label 2-135-9.4-c3-0-5
Degree $2$
Conductor $135$
Sign $-0.320 - 0.947i$
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  + (1.87 + 3.24i)2-s + (−3.02 + 5.24i)4-s + (2.5 − 4.33i)5-s + (15.6 + 27.1i)7-s + 7.30·8-s + 18.7·10-s + (−10.4 − 18.0i)11-s + (−29.9 + 51.9i)13-s + (−58.7 + 101. i)14-s + (37.8 + 65.6i)16-s + 74.0·17-s − 63.8·19-s + (15.1 + 26.2i)20-s + (39.0 − 67.6i)22-s + (−16.4 + 28.4i)23-s + ⋯
L(s)  = 1  + (0.662 + 1.14i)2-s + (−0.378 + 0.655i)4-s + (0.223 − 0.387i)5-s + (0.846 + 1.46i)7-s + 0.322·8-s + 0.592·10-s + (−0.285 − 0.494i)11-s + (−0.639 + 1.10i)13-s + (−1.12 + 1.94i)14-s + (0.592 + 1.02i)16-s + 1.05·17-s − 0.770·19-s + (0.169 + 0.292i)20-s + (0.378 − 0.655i)22-s + (−0.148 + 0.257i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.320 - 0.947i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.50924 + 2.10284i\)
\(L(\frac12)\) \(\approx\) \(1.50924 + 2.10284i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.87 - 3.24i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-15.6 - 27.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (10.4 + 18.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (29.9 - 51.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 74.0T + 4.91e3T^{2} \)
19 \( 1 + 63.8T + 6.85e3T^{2} \)
23 \( 1 + (16.4 - 28.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (80.0 + 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-127. + 220. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 215.T + 5.06e4T^{2} \)
41 \( 1 + (-70.8 + 122. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (68.9 + 119. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-16.7 - 29.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 41.9T + 1.48e5T^{2} \)
59 \( 1 + (-307. + 532. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-67.1 - 116. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-428. + 742. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 588.T + 3.57e5T^{2} \)
73 \( 1 + 618.T + 3.89e5T^{2} \)
79 \( 1 + (-172. - 299. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (546. + 946. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 414.T + 7.04e5T^{2} \)
97 \( 1 + (-100. - 174. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.31383876902124408640795723498, −12.21124245169516807805841753700, −11.33283188424833521411027682085, −9.701527106169209683727949124486, −8.519680275820052189551838842321, −7.68690004819193143328026107302, −6.15916798433336867799682743872, −5.45216812268642077214889883797, −4.40490300439431095192658962542, −2.09778957645580124203290300113, 1.23716402281219157545915361423, 2.84651664562852260055261528008, 4.16676995840578953738506014239, 5.20819166132410274418430423085, 7.16851289914727623220899198667, 7.988505398999070578615265053809, 10.16148847513077456914455812931, 10.38434220892771702833807433174, 11.38710322604891576196009644635, 12.52529529521287056338510735701

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