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Properties

Label 2-135-45.22-c2-0-7
Degree $2$
Conductor $135$
Sign $-0.278 + 0.960i$
Analytic cond. $3.67848$
Root an. cond. $1.91793$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.644 − 2.40i)2-s + (−1.90 − 1.10i)4-s + (4.49 − 2.18i)5-s + (0.0176 − 0.0658i)7-s + (3.16 − 3.16i)8-s + (−2.34 − 12.2i)10-s + (−4.62 − 8.00i)11-s + (3.38 + 12.6i)13-s + (−0.146 − 0.0848i)14-s + (−9.97 − 17.2i)16-s + (−13.6 − 13.6i)17-s + 5.80i·19-s + (−10.9 − 0.794i)20-s + (−22.2 + 5.96i)22-s + (11.7 + 43.8i)23-s + ⋯
L(s)  = 1  + (0.322 − 1.20i)2-s + (−0.477 − 0.275i)4-s + (0.899 − 0.436i)5-s + (0.00251 − 0.00940i)7-s + (0.395 − 0.395i)8-s + (−0.234 − 1.22i)10-s + (−0.420 − 0.727i)11-s + (0.260 + 0.972i)13-s + (−0.0104 − 0.00606i)14-s + (−0.623 − 1.08i)16-s + (−0.800 − 0.800i)17-s + 0.305i·19-s + (−0.549 − 0.0397i)20-s + (−1.01 + 0.270i)22-s + (0.510 + 1.90i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.278 + 0.960i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ -0.278 + 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.16562 - 1.55178i\)
\(L(\frac12)\) \(\approx\) \(1.16562 - 1.55178i\)
\(L(2)\) 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 \( 1 + (-4.49 + 2.18i)T \)
good2 \( 1 + (-0.644 + 2.40i)T + (-3.46 - 2i)T^{2} \)
7 \( 1 + (-0.0176 + 0.0658i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (4.62 + 8.00i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.38 - 12.6i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (13.6 + 13.6i)T + 289iT^{2} \)
19 \( 1 - 5.80iT - 361T^{2} \)
23 \( 1 + (-11.7 - 43.8i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (6.37 - 3.67i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (8.35 - 14.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-36.0 - 36.0i)T + 1.36e3iT^{2} \)
41 \( 1 + (-31.1 + 53.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-9.70 - 2.59i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (11.9 - 44.6i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (28.0 - 28.0i)T - 2.80e3iT^{2} \)
59 \( 1 + (16.3 + 9.44i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-2.69 - 4.66i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (47.9 - 12.8i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 56.4T + 5.04e3T^{2} \)
73 \( 1 + (54.7 - 54.7i)T - 5.32e3iT^{2} \)
79 \( 1 + (40.9 - 23.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (60.0 + 16.0i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 59.1iT - 7.92e3T^{2} \)
97 \( 1 + (-40.2 + 150. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.71258174029503489122869305104, −11.56806883425323937816114676908, −10.90529747896259219604466602004, −9.697770702661819362138668475953, −8.955641002483911845792676975659, −7.23265887959302128918330734649, −5.75324148215518997798622254833, −4.42306808214168123903640205098, −2.88211039400084390810079839480, −1.46253345165526566099654047940, 2.36867494023156121256475332492, 4.59311083803556636757658618724, 5.79747800442183257194916944782, 6.60422350807922805447805454510, 7.68836665660795959855123612804, 8.855758451256544083191558738786, 10.28638598521414778578713816219, 10.98074359725074205263364154725, 12.86391669276551455175652917756, 13.33087096629515630194895220238

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