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Properties

Label 2-135-5.3-c2-0-5
Degree $2$
Conductor $135$
Sign $0.300 - 0.953i$
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  + (−2.69 + 2.69i)2-s − 10.5i·4-s + (−4.29 − 2.55i)5-s + (−1.13 + 1.13i)7-s + (17.5 + 17.5i)8-s + (18.4 − 4.67i)10-s + 9.92·11-s + (14.1 + 14.1i)13-s − 6.13i·14-s − 52.5·16-s + (6.77 − 6.77i)17-s − 4.72i·19-s + (−26.9 + 45.1i)20-s + (−26.7 + 26.7i)22-s + (10.0 + 10.0i)23-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s − 2.62i·4-s + (−0.858 − 0.511i)5-s + (−0.162 + 0.162i)7-s + (2.19 + 2.19i)8-s + (1.84 − 0.467i)10-s + 0.902·11-s + (1.09 + 1.09i)13-s − 0.438i·14-s − 3.28·16-s + (0.398 − 0.398i)17-s − 0.248i·19-s + (−1.34 + 2.25i)20-s + (−1.21 + 1.21i)22-s + (0.437 + 0.437i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ 0.300 - 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.519038 + 0.380477i\)
\(L(\frac12)\) \(\approx\) \(0.519038 + 0.380477i\)
\(L(2)\) 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 \( 1 + (4.29 + 2.55i)T \)
good2 \( 1 + (2.69 - 2.69i)T - 4iT^{2} \)
7 \( 1 + (1.13 - 1.13i)T - 49iT^{2} \)
11 \( 1 - 9.92T + 121T^{2} \)
13 \( 1 + (-14.1 - 14.1i)T + 169iT^{2} \)
17 \( 1 + (-6.77 + 6.77i)T - 289iT^{2} \)
19 \( 1 + 4.72iT - 361T^{2} \)
23 \( 1 + (-10.0 - 10.0i)T + 529iT^{2} \)
29 \( 1 + 20.1iT - 841T^{2} \)
31 \( 1 - 34.8T + 961T^{2} \)
37 \( 1 + (-29.5 + 29.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.7T + 1.68e3T^{2} \)
43 \( 1 + (15.3 + 15.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (15.4 - 15.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-44.2 - 44.2i)T + 2.80e3iT^{2} \)
59 \( 1 + 7.95iT - 3.48e3T^{2} \)
61 \( 1 - 10.1T + 3.72e3T^{2} \)
67 \( 1 + (41.6 - 41.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 100.T + 5.04e3T^{2} \)
73 \( 1 + (91.7 + 91.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 8.36iT - 6.24e3T^{2} \)
83 \( 1 + (-30.7 - 30.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 87.6iT - 7.92e3T^{2} \)
97 \( 1 + (20.4 - 20.4i)T - 9.40e3iT^{2} \)
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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

−13.58748998142768466209201056962, −11.84478998686448162136753345178, −10.98053439665303524118677113675, −9.461306804789706066953561597346, −8.938687596098808726504566040718, −7.942342843475525651728305760092, −6.92338606295414756010067445186, −5.91965161470818562430298324604, −4.35996770266935475188151995997, −1.06508265008588356624278025718, 0.973891544322046377403104996556, 3.03776627442283308848030150588, 3.94798752958453444839580266392, 6.68892724217384854642037313473, 7.976703921622503958524121023661, 8.605951873028550391831585399214, 9.925917822214968534691158938929, 10.72643065057004676520972883745, 11.49092864071539868800950095469, 12.33852127236463759794512190822

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