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Properties

Label 2-135-9.4-c1-0-2
Degree $2$
Conductor $135$
Sign $0.983 + 0.182i$
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  + (0.285 + 0.495i)2-s + (0.836 − 1.44i)4-s + (0.5 − 0.866i)5-s + (−0.714 − 1.23i)7-s + 2.10·8-s + 0.571·10-s + (1.33 + 2.31i)11-s + (−2.33 + 4.04i)13-s + (0.408 − 0.707i)14-s + (−1.07 − 1.85i)16-s − 2.67·17-s + 4.67·19-s + (−0.836 − 1.44i)20-s + (−0.764 + 1.32i)22-s + (−2.95 + 5.12i)23-s + ⋯
L(s)  = 1  + (0.202 + 0.350i)2-s + (0.418 − 0.724i)4-s + (0.223 − 0.387i)5-s + (−0.269 − 0.467i)7-s + 0.742·8-s + 0.180·10-s + (0.402 + 0.697i)11-s + (−0.648 + 1.12i)13-s + (0.109 − 0.189i)14-s + (−0.267 − 0.464i)16-s − 0.648·17-s + 1.07·19-s + (−0.187 − 0.323i)20-s + (−0.162 + 0.282i)22-s + (−0.616 + 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.983 + 0.182i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ 0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31006 - 0.120329i\)
\(L(\frac12)\) \(\approx\) \(1.31006 - 0.120329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.285 - 0.495i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.714 + 1.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.33 - 4.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + (2.95 - 5.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.74 + 8.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + (0.735 - 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.235 + 0.408i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.47 - 6.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.14T + 53T^{2} \)
59 \( 1 + (0.571 - 0.990i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.29 + 5.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 + (-0.143 - 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.14 + 3.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (3.91 + 6.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.52963791320791963007931768755, −12.13191932357697983638799054452, −11.23767579327843315159077816066, −9.895300513086888562392997161230, −9.349655172260991802716175843207, −7.50344479865651586261002657818, −6.70588699635044621079410971689, −5.42861301970256754403482849925, −4.22357116927821807749007395439, −1.84135831178399842354635387809, 2.50066657728374814554970133046, 3.63996235304626027350577831368, 5.46225681079787195220808514930, 6.79965125248867165690370437399, 7.890624056510962625192394053316, 9.094003964434575096653067938533, 10.41742638563194328672600054966, 11.30653482789878423163023681659, 12.29380614496135748726612790762, 13.05280537356610781137803163313

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