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Properties

Label 2-135-9.7-c3-0-3
Degree $2$
Conductor $135$
Sign $-0.314 - 0.949i$
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  + (−0.0874 + 0.151i)2-s + (3.98 + 6.90i)4-s + (2.5 + 4.33i)5-s + (−4.23 + 7.32i)7-s − 2.79·8-s − 0.874·10-s + (−15.7 + 27.2i)11-s + (−13.4 − 23.2i)13-s + (−0.740 − 1.28i)14-s + (−31.6 + 54.7i)16-s + 44.3·17-s − 90.2·19-s + (−19.9 + 34.5i)20-s + (−2.75 − 4.77i)22-s + (97.1 + 168. i)23-s + ⋯
L(s)  = 1  + (−0.0309 + 0.0535i)2-s + (0.498 + 0.862i)4-s + (0.223 + 0.387i)5-s + (−0.228 + 0.395i)7-s − 0.123·8-s − 0.0276·10-s + (−0.431 + 0.747i)11-s + (−0.286 − 0.496i)13-s + (−0.0141 − 0.0244i)14-s + (−0.494 + 0.856i)16-s + 0.632·17-s − 1.08·19-s + (−0.222 + 0.385i)20-s + (−0.0267 − 0.0462i)22-s + (0.880 + 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.314 - 0.949i$
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: \((2,\ 135,\ (\ :3/2),\ -0.314 - 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.906511 + 1.25526i\)
\(L(\frac12)\) \(\approx\) \(0.906511 + 1.25526i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.0874 - 0.151i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (4.23 - 7.32i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (15.7 - 27.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (13.4 + 23.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 44.3T + 4.91e3T^{2} \)
19 \( 1 + 90.2T + 6.85e3T^{2} \)
23 \( 1 + (-97.1 - 168. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-1.87 + 3.24i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-125. - 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 62.2T + 5.06e4T^{2} \)
41 \( 1 + (102. + 176. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-263. + 456. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-77.8 + 134. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 141.T + 1.48e5T^{2} \)
59 \( 1 + (246. + 427. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-379. + 657. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-271. - 470. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 928.T + 3.57e5T^{2} \)
73 \( 1 - 608.T + 3.89e5T^{2} \)
79 \( 1 + (307. - 532. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-537. + 931. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (-166. + 288. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.77856845898799671875217343104, −12.25925440616563056498797317128, −11.04241169592774888521381277365, −10.04999864422675850294296980877, −8.768918762595870184465219340605, −7.60592691694319254157811626649, −6.74140541695804851450080587645, −5.30936765784730685166411725727, −3.51366797203490508322428768076, −2.27852418771131265195829680952, 0.77680866648430170685307633238, 2.54482508710635078438093378137, 4.53586339337128388306681045214, 5.85390930300282892862629358172, 6.79322936238498789657160186620, 8.261014064583771191548522566948, 9.488455504890579781438066181283, 10.43797841932472924376274975575, 11.23626863485155037540042821678, 12.47628872471223685385056120403

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