Properties

Label 2-135-9.7-c3-0-6
Degree $2$
Conductor $135$
Sign $0.494 - 0.868i$
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  + (−2.28 + 3.96i)2-s + (−6.45 − 11.1i)4-s + (2.5 + 4.33i)5-s + (10.0 − 17.4i)7-s + 22.4·8-s − 22.8·10-s + (33.1 − 57.4i)11-s + (23.4 + 40.5i)13-s + (45.9 + 79.6i)14-s + (0.237 − 0.411i)16-s + 47.6·17-s − 9.95·19-s + (32.2 − 55.9i)20-s + (151. + 262. i)22-s + (4.79 + 8.30i)23-s + ⋯
L(s)  = 1  + (−0.808 + 1.40i)2-s + (−0.807 − 1.39i)4-s + (0.223 + 0.387i)5-s + (0.543 − 0.940i)7-s + 0.993·8-s − 0.723·10-s + (0.909 − 1.57i)11-s + (0.499 + 0.864i)13-s + (0.878 + 1.52i)14-s + (0.00371 − 0.00643i)16-s + 0.679·17-s − 0.120·19-s + (0.361 − 0.625i)20-s + (1.47 + 2.54i)22-s + (0.0434 + 0.0753i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.494 - 0.868i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ 0.494 - 0.868i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01257 + 0.588571i\)
\(L(\frac12)\) \(\approx\) \(1.01257 + 0.588571i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.28 - 3.96i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-10.0 + 17.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-33.1 + 57.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-23.4 - 40.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 47.6T + 4.91e3T^{2} \)
19 \( 1 + 9.95T + 6.85e3T^{2} \)
23 \( 1 + (-4.79 - 8.30i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (89.3 - 154. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (77.0 + 133. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 248.T + 5.06e4T^{2} \)
41 \( 1 + (-124. - 216. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-106. + 183. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-237. + 411. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 546.T + 1.48e5T^{2} \)
59 \( 1 + (209. + 363. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-272. + 472. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-223. - 387. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 409.T + 3.57e5T^{2} \)
73 \( 1 + 358.T + 3.89e5T^{2} \)
79 \( 1 + (-325. + 564. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (406. - 704. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 200.T + 7.04e5T^{2} \)
97 \( 1 + (126. - 218. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.50792903439107953419524670005, −11.57622447919854887769206997700, −10.71005862134710269434846349672, −9.426001286079479951448172437398, −8.566803923803790250362758621263, −7.52146959219496767034201757852, −6.57032579213853760814771673729, −5.63782406773366193130652842118, −3.82378101477266373673975012897, −0.982715769674182569809337261981, 1.28117760485720698467757558440, 2.46606726067984353045417699574, 4.17291353941390207178393797955, 5.75205965696564852229317901696, 7.68194773378565401510079883432, 8.821984139012401475586786816176, 9.523437610370136138452870100445, 10.46952364991450494798308548235, 11.63400386630990849680060850315, 12.29049981315727158511076891642

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