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Properties

Label 2-135-9.7-c3-0-4
Degree $2$
Conductor $135$
Sign $-0.369 - 0.929i$
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  + (−2.65 + 4.60i)2-s + (−10.1 − 17.5i)4-s + (−2.5 − 4.33i)5-s + (6.71 − 11.6i)7-s + 65.1·8-s + 26.5·10-s + (−23.4 + 40.6i)11-s + (18.0 + 31.2i)13-s + (35.7 + 61.8i)14-s + (−92.1 + 159. i)16-s + 54.6·17-s + 111.·19-s + (−50.6 + 87.7i)20-s + (−124. − 216. i)22-s + (−17.9 − 31.1i)23-s + ⋯
L(s)  = 1  + (−0.939 + 1.62i)2-s + (−1.26 − 2.19i)4-s + (−0.223 − 0.387i)5-s + (0.362 − 0.628i)7-s + 2.87·8-s + 0.840·10-s + (−0.643 + 1.11i)11-s + (0.385 + 0.667i)13-s + (0.681 + 1.18i)14-s + (−1.43 + 2.49i)16-s + 0.779·17-s + 1.34·19-s + (−0.566 + 0.980i)20-s + (−1.20 − 2.09i)22-s + (−0.163 − 0.282i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.369 - 0.929i$
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: \((2,\ 135,\ (\ :3/2),\ -0.369 - 0.929i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.496714 + 0.732145i\)
\(L(\frac12)\) \(\approx\) \(0.496714 + 0.732145i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.65 - 4.60i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-6.71 + 11.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (23.4 - 40.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-18.0 - 31.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 54.6T + 4.91e3T^{2} \)
19 \( 1 - 111.T + 6.85e3T^{2} \)
23 \( 1 + (17.9 + 31.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-29.0 + 50.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-147. - 256. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 53.0T + 5.06e4T^{2} \)
41 \( 1 + (-64.1 - 111. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-82.0 + 142. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-43.9 + 76.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 479.T + 1.48e5T^{2} \)
59 \( 1 + (-317. - 550. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (24.0 - 41.5i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-14.4 - 25.0i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 576.T + 3.57e5T^{2} \)
73 \( 1 - 835.T + 3.89e5T^{2} \)
79 \( 1 + (-101. + 176. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-232. + 402. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 993.T + 7.04e5T^{2} \)
97 \( 1 + (440. - 763. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.59785303123792501910942890981, −12.06644870177002265034005291739, −10.47465547074526446814791876081, −9.711124658223245043101850229386, −8.595219637283440458546453405555, −7.62957250230937358046685861963, −6.96667566028636450601273294767, −5.49167016148099000206547825072, −4.48724419314214330526548208483, −1.11712603820026978086191516993, 0.813295609616942312595728779940, 2.62340949402744693101903220446, 3.57975621566429262002160639109, 5.50019329828926500891825499310, 7.74844206955221524873771905960, 8.379652967502441982720120918052, 9.558679105451514003292913030034, 10.49486953417285907565692704840, 11.36709715302878731554741870072, 11.98289126304072316022489498608

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