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Properties

Label 2-135-5.3-c2-0-1
Degree $2$
Conductor $135$
Sign $-0.675 + 0.737i$
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  + (−2.52 + 2.52i)2-s − 8.73i·4-s + (2.81 + 4.13i)5-s + (−4.39 + 4.39i)7-s + (11.9 + 11.9i)8-s + (−17.5 − 3.32i)10-s − 20.6·11-s + (4.07 + 4.07i)13-s − 22.1i·14-s − 25.3·16-s + (−3.85 + 3.85i)17-s − 18.4i·19-s + (36.0 − 24.5i)20-s + (52.1 − 52.1i)22-s + (−14.1 − 14.1i)23-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)2-s − 2.18i·4-s + (0.562 + 0.826i)5-s + (−0.627 + 0.627i)7-s + (1.49 + 1.49i)8-s + (−1.75 − 0.332i)10-s − 1.87·11-s + (0.313 + 0.313i)13-s − 1.58i·14-s − 1.58·16-s + (−0.226 + 0.226i)17-s − 0.973i·19-s + (1.80 − 1.22i)20-s + (2.36 − 2.36i)22-s + (−0.616 − 0.616i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.675 + 0.737i$
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: \((2,\ 135,\ (\ :1),\ -0.675 + 0.737i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.124802 - 0.283418i\)
\(L(\frac12)\) \(\approx\) \(0.124802 - 0.283418i\)
\(L(2)\) 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 \( 1 + (-2.81 - 4.13i)T \)
good2 \( 1 + (2.52 - 2.52i)T - 4iT^{2} \)
7 \( 1 + (4.39 - 4.39i)T - 49iT^{2} \)
11 \( 1 + 20.6T + 121T^{2} \)
13 \( 1 + (-4.07 - 4.07i)T + 169iT^{2} \)
17 \( 1 + (3.85 - 3.85i)T - 289iT^{2} \)
19 \( 1 + 18.4iT - 361T^{2} \)
23 \( 1 + (14.1 + 14.1i)T + 529iT^{2} \)
29 \( 1 + 10.6iT - 841T^{2} \)
31 \( 1 + 33.1T + 961T^{2} \)
37 \( 1 + (35.8 - 35.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 12.0T + 1.68e3T^{2} \)
43 \( 1 + (-35.5 - 35.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-0.615 + 0.615i)T - 2.20e3iT^{2} \)
53 \( 1 + (11.6 + 11.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 84.1iT - 3.48e3T^{2} \)
61 \( 1 - 69.7T + 3.72e3T^{2} \)
67 \( 1 + (38.4 - 38.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 107.T + 5.04e3T^{2} \)
73 \( 1 + (-8.17 - 8.17i)T + 5.32e3iT^{2} \)
79 \( 1 - 73.0iT - 6.24e3T^{2} \)
83 \( 1 + (-4.18 - 4.18i)T + 6.88e3iT^{2} \)
89 \( 1 - 6.20iT - 7.92e3T^{2} \)
97 \( 1 + (54.9 - 54.9i)T - 9.40e3iT^{2} \)
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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

−13.91044979522921348134238303695, −12.89189639981086695984474225587, −11.00082351720150452189467538548, −10.23091474253900109281229331349, −9.359880444560721607017238637462, −8.314442197119300594846776340372, −7.22685191411195929207296382648, −6.27186967541212099971694846115, −5.37899174572234123771828927485, −2.51449114947540383011617269697, 0.29082816366295675344015228375, 2.05411414511962894359348485198, 3.59482367407086232725461886368, 5.48844265001962931647975901654, 7.50405124980618353242425574793, 8.387102666745448797834075789243, 9.488550838045796098764225118360, 10.25655690319591938819593990038, 10.90612873558398611381656604271, 12.40677357798177124916121087232

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