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Properties

Label 2-135-9.4-c3-0-11
Degree $2$
Conductor $135$
Sign $-0.925 - 0.379i$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09 − 1.90i)2-s + (1.59 − 2.75i)4-s + (−2.5 + 4.33i)5-s + (−1.38 − 2.39i)7-s − 24.5·8-s + 10.9·10-s + (−26.3 − 45.6i)11-s + (−10.2 + 17.7i)13-s + (−3.03 + 5.25i)14-s + (14.1 + 24.5i)16-s − 3.66·17-s − 95.6·19-s + (7.96 + 13.7i)20-s + (−57.7 + 100. i)22-s + (−44.9 + 77.8i)23-s + ⋯
L(s)  = 1  + (−0.387 − 0.671i)2-s + (0.199 − 0.344i)4-s + (−0.223 + 0.387i)5-s + (−0.0746 − 0.129i)7-s − 1.08·8-s + 0.346·10-s + (−0.721 − 1.25i)11-s + (−0.218 + 0.377i)13-s + (−0.0579 + 0.100i)14-s + (0.221 + 0.384i)16-s − 0.0522·17-s − 1.15·19-s + (0.0890 + 0.154i)20-s + (−0.559 + 0.969i)22-s + (−0.407 + 0.705i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.925 - 0.379i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0877754 + 0.445878i\)
\(L(\frac12)\) \(\approx\) \(0.0877754 + 0.445878i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T \)
good2 \( 1 + (1.09 + 1.90i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (1.38 + 2.39i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (26.3 + 45.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (10.2 - 17.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 3.66T + 4.91e3T^{2} \)
19 \( 1 + 95.6T + 6.85e3T^{2} \)
23 \( 1 + (44.9 - 77.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (113. + 197. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (139. - 241. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 273.T + 5.06e4T^{2} \)
41 \( 1 + (-32.4 + 56.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (209. + 362. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-69.3 - 120. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 197.T + 1.48e5T^{2} \)
59 \( 1 + (-370. + 641. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (244. + 423. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-205. + 356. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 310.T + 3.57e5T^{2} \)
73 \( 1 + 51.0T + 3.89e5T^{2} \)
79 \( 1 + (603. + 1.04e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-452. - 783. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 663.T + 7.04e5T^{2} \)
97 \( 1 + (-362. - 628. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.85334296939535226529876871229, −11.03045564776922404609952815724, −10.36526975243456628318439475460, −9.219323122167660314854379065028, −8.087367528854900499850672536024, −6.63394909361432494815263268925, −5.53814586225030760765813041283, −3.59254523932058990655071436996, −2.19332379383437871326068566410, −0.23781164905914572162973718435, 2.46937174346943544139808879212, 4.30221169623915700219859371669, 5.79016675697994731632366962364, 7.10962378337042582636004795769, 7.917416317053613223252499089724, 8.918865517017759968222932675096, 10.04534457524045889395273489194, 11.34889519371119248205060865157, 12.60490505007730128503389064556, 12.91130525888238689857598106056

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