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Properties

Label 2-135-45.2-c1-0-1
Degree $2$
Conductor $135$
Sign $0.224 - 0.974i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.430 + 1.60i)2-s + (−0.661 + 0.382i)4-s + (2.23 − 0.154i)5-s + (−1.73 + 0.465i)7-s + (1.45 + 1.45i)8-s + (1.20 + 3.51i)10-s + (−3.12 − 1.80i)11-s + (−1.27 − 0.342i)13-s + (−1.49 − 2.59i)14-s + (−2.47 + 4.28i)16-s + (−0.277 + 0.277i)17-s − 6.25i·19-s + (−1.41 + 0.954i)20-s + (1.55 − 5.79i)22-s + (−0.579 + 2.16i)23-s + ⋯
L(s)  = 1  + (0.304 + 1.13i)2-s + (−0.330 + 0.191i)4-s + (0.997 − 0.0690i)5-s + (−0.656 + 0.175i)7-s + (0.513 + 0.513i)8-s + (0.381 + 1.11i)10-s + (−0.942 − 0.544i)11-s + (−0.354 − 0.0950i)13-s + (−0.399 − 0.692i)14-s + (−0.618 + 1.07i)16-s + (−0.0671 + 0.0671i)17-s − 1.43i·19-s + (−0.317 + 0.213i)20-s + (0.331 − 1.23i)22-s + (−0.120 + 0.451i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ 0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08984 + 0.867677i\)
\(L(\frac12)\) \(\approx\) \(1.08984 + 0.867677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.154i)T \)
good2 \( 1 + (-0.430 - 1.60i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (1.73 - 0.465i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.12 + 1.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.27 + 0.342i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.277 - 0.277i)T - 17iT^{2} \)
19 \( 1 + 6.25iT - 19T^{2} \)
23 \( 1 + (0.579 - 2.16i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.55 - 5.55i)T + 37iT^{2} \)
41 \( 1 + (1.29 - 0.744i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.10 + 4.10i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.02 - 3.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.48 + 7.48i)T + 53iT^{2} \)
59 \( 1 + (0.279 + 0.483i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.90 - 10.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.01iT - 71T^{2} \)
73 \( 1 + (1.29 - 1.29i)T - 73iT^{2} \)
79 \( 1 + (-6.96 - 4.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.560 + 0.150i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (-5.14 + 1.37i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.39752001288849070650986663323, −13.10128066730509068663681281249, −11.35268645664437463601775033846, −10.23549369008310889075561701017, −9.162869206228958124951902783923, −7.919536849309352776090902931203, −6.71260258218349287162214239932, −5.84854460353841370158287952471, −4.90186971677284377185300052300, −2.60702084255707795170750880423, 1.97784256377462236875226182694, 3.24174181441643058254016653248, 4.85895122913479755197422288863, 6.31937988713544436864173471379, 7.60326206560354679500640889595, 9.370660872150744853535878452655, 10.19656769640363755727973001479, 10.76572430816459708344671834440, 12.29706633127689081020911612545, 12.76797068094651668194901597232

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