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Properties

Label 2-135-45.14-c2-0-5
Degree $2$
Conductor $135$
Sign $0.717 - 0.696i$
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  + (1.14 + 1.98i)2-s + (−0.629 + 1.09i)4-s + (3.96 − 3.05i)5-s + (6.04 − 3.49i)7-s + 6.28·8-s + (10.6 + 4.36i)10-s + (−15.8 + 9.12i)11-s + (3.66 + 2.11i)13-s + (13.8 + 8.00i)14-s + (9.72 + 16.8i)16-s − 17.3·17-s − 3.96·19-s + (0.834 + 6.24i)20-s + (−36.2 − 20.9i)22-s + (0.287 − 0.498i)23-s + ⋯
L(s)  = 1  + (0.573 + 0.993i)2-s + (−0.157 + 0.272i)4-s + (0.792 − 0.610i)5-s + (0.863 − 0.498i)7-s + 0.785·8-s + (1.06 + 0.436i)10-s + (−1.43 + 0.829i)11-s + (0.282 + 0.162i)13-s + (0.990 + 0.571i)14-s + (0.607 + 1.05i)16-s − 1.01·17-s − 0.208·19-s + (0.0417 + 0.312i)20-s + (−1.64 − 0.951i)22-s + (0.0125 − 0.0216i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.717 - 0.696i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ 0.717 - 0.696i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.07886 + 0.842646i\)
\(L(\frac12)\) \(\approx\) \(2.07886 + 0.842646i\)
\(L(2)\) 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 \( 1 + (-3.96 + 3.05i)T \)
good2 \( 1 + (-1.14 - 1.98i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (-6.04 + 3.49i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.8 - 9.12i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.66 - 2.11i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 17.3T + 289T^{2} \)
19 \( 1 + 3.96T + 361T^{2} \)
23 \( 1 + (-0.287 + 0.498i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-18.1 + 10.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (16.7 - 28.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 21.4iT - 1.36e3T^{2} \)
41 \( 1 + (44.1 + 25.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-7.15 + 4.13i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (7.57 + 13.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 24.5T + 2.80e3T^{2} \)
59 \( 1 + (43.1 + 24.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (31.4 + 54.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (103. + 59.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 66.8iT - 5.04e3T^{2} \)
73 \( 1 + 48.9iT - 5.32e3T^{2} \)
79 \( 1 + (-58.9 - 102. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-3.66 - 6.34i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + (-3.59 + 2.07i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.46721329942435408318059931478, −12.53889294698879001192779066659, −10.88926357904639986854695146811, −10.13916821263200186013444424245, −8.598630019587182461109954711079, −7.59240854765519568660717287855, −6.47052212855235673139635546683, −5.15635535112922021897034976638, −4.59770807688905038209471814977, −1.88018511403018651169563390820, 2.03185802816867467261477560952, 3.04586875289586781336760260863, 4.79670598046287803088159155256, 5.91005103772105615915283481462, 7.55640587961795482487374437071, 8.737917864203809781113156235278, 10.32433390346014173951837922701, 10.90678605555060440730391509969, 11.69204366416130128947569540778, 13.07710517997731796592604253310

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