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Properties

Label 2-135-9.2-c2-0-4
Degree $2$
Conductor $135$
Sign $0.158 - 0.987i$
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  + (3.23 + 1.86i)2-s + (4.98 + 8.62i)4-s + (−1.93 + 1.11i)5-s + (3.63 − 6.28i)7-s + 22.2i·8-s − 8.35·10-s + (−6.50 − 3.75i)11-s + (−1.46 − 2.54i)13-s + (23.4 − 13.5i)14-s + (−21.6 + 37.5i)16-s − 1.90i·17-s − 7.38·19-s + (−19.2 − 11.1i)20-s + (−14.0 − 24.3i)22-s + (30.6 − 17.6i)23-s + ⋯
L(s)  = 1  + (1.61 + 0.934i)2-s + (1.24 + 2.15i)4-s + (−0.387 + 0.223i)5-s + (0.518 − 0.898i)7-s + 2.78i·8-s − 0.835·10-s + (−0.591 − 0.341i)11-s + (−0.112 − 0.195i)13-s + (1.67 − 0.968i)14-s + (−1.35 + 2.34i)16-s − 0.111i·17-s − 0.388·19-s + (−0.964 − 0.556i)20-s + (−0.638 − 1.10i)22-s + (1.33 − 0.769i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.158 - 0.987i$
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: \((2,\ 135,\ (\ :1),\ 0.158 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.38585 + 2.03390i\)
\(L(\frac12)\) \(\approx\) \(2.38585 + 2.03390i\)
\(L(2)\) 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 \( 1 + (-3.23 - 1.86i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (-3.63 + 6.28i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (6.50 + 3.75i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1.46 + 2.54i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 1.90iT - 289T^{2} \)
19 \( 1 + 7.38T + 361T^{2} \)
23 \( 1 + (-30.6 + 17.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-14.2 - 8.19i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (13.3 + 23.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 44.6T + 1.36e3T^{2} \)
41 \( 1 + (14.8 - 8.58i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (20.5 - 35.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (36.4 + 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-4.12 + 2.37i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-38.4 + 66.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-41.9 - 72.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 23.1iT - 5.04e3T^{2} \)
73 \( 1 + 103.T + 5.32e3T^{2} \)
79 \( 1 + (40.2 - 69.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-17.1 - 9.87i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 29.0iT - 7.92e3T^{2} \)
97 \( 1 + (-47.7 + 82.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.37602610443316663759268444728, −12.62520139310106027966539632060, −11.44990288080127825365147722396, −10.60499251977814991991756400724, −8.410087459560668522915732889281, −7.47714653702140675085170799163, −6.63142468990725623049450022457, −5.23443209178105598204802596864, −4.28772249014205613768762571943, −3.01185415457000682919306497291, 1.91921615621147888533490220197, 3.31681292962069824385369404637, 4.78717252688856100046845412192, 5.46147627138319959708714239518, 6.95784085549597339274966932883, 8.669600318787526653400560317573, 10.13530078277705702993546625172, 11.18584561193245723467629401824, 11.92631704150092193954938416511, 12.69223536069959177039757191557

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