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Properties

Label 2-135-5.3-c2-0-10
Degree $2$
Conductor $135$
Sign $0.955 + 0.293i$
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  + (0.683 − 0.683i)2-s + 3.06i·4-s + (2.52 − 4.31i)5-s + (1.91 − 1.91i)7-s + (4.82 + 4.82i)8-s + (−1.22 − 4.67i)10-s + 16.0·11-s + (0.607 + 0.607i)13-s − 2.61i·14-s − 5.66·16-s + (7.56 − 7.56i)17-s + 21.0i·19-s + (13.2 + 7.75i)20-s + (10.9 − 10.9i)22-s + (−25.2 − 25.2i)23-s + ⋯
L(s)  = 1  + (0.341 − 0.341i)2-s + 0.766i·4-s + (0.505 − 0.862i)5-s + (0.273 − 0.273i)7-s + (0.603 + 0.603i)8-s + (−0.122 − 0.467i)10-s + 1.45·11-s + (0.0467 + 0.0467i)13-s − 0.186i·14-s − 0.354·16-s + (0.444 − 0.444i)17-s + 1.10i·19-s + (0.661 + 0.387i)20-s + (0.498 − 0.498i)22-s + (−1.09 − 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.955 + 0.293i$
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: \((2,\ 135,\ (\ :1),\ 0.955 + 0.293i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90748 - 0.286569i\)
\(L(\frac12)\) \(\approx\) \(1.90748 - 0.286569i\)
\(L(2)\) 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 \( 1 + (-2.52 + 4.31i)T \)
good2 \( 1 + (-0.683 + 0.683i)T - 4iT^{2} \)
7 \( 1 + (-1.91 + 1.91i)T - 49iT^{2} \)
11 \( 1 - 16.0T + 121T^{2} \)
13 \( 1 + (-0.607 - 0.607i)T + 169iT^{2} \)
17 \( 1 + (-7.56 + 7.56i)T - 289iT^{2} \)
19 \( 1 - 21.0iT - 361T^{2} \)
23 \( 1 + (25.2 + 25.2i)T + 529iT^{2} \)
29 \( 1 - 13.4iT - 841T^{2} \)
31 \( 1 + 39.2T + 961T^{2} \)
37 \( 1 + (-5.47 + 5.47i)T - 1.36e3iT^{2} \)
41 \( 1 + 69.4T + 1.68e3T^{2} \)
43 \( 1 + (-55.6 - 55.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (13.7 - 13.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (34.8 + 34.8i)T + 2.80e3iT^{2} \)
59 \( 1 + 62.0iT - 3.48e3T^{2} \)
61 \( 1 + 67.2T + 3.72e3T^{2} \)
67 \( 1 + (-29.2 + 29.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 71.3T + 5.04e3T^{2} \)
73 \( 1 + (23.5 + 23.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 5.61iT - 6.24e3T^{2} \)
83 \( 1 + (-26.0 - 26.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 23.2iT - 7.92e3T^{2} \)
97 \( 1 + (-97.3 + 97.3i)T - 9.40e3iT^{2} \)
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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

−12.65989427264040582727244935294, −12.21702540029298814653338631435, −11.16826526884953219423604346204, −9.768981363768056182570326038907, −8.731718943980329088464113166810, −7.75127512199413510763391087251, −6.25217094871990692333352292242, −4.73795795656719340825586667347, −3.69057896592849112807019092204, −1.70443327921127932663742350260, 1.77624366969302349762740980827, 3.83059846785775972340686148268, 5.45260579217728344680297132688, 6.35350777866979293779972431033, 7.30080708941317440446184223034, 9.069515401360855779735588216425, 9.937471620444605498403724980880, 10.98100003933627255786507729806, 11.90804851676726275721938854721, 13.46846254104812092368078213808

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