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Properties

Label 2-108-36.11-c3-0-1
Degree $2$
Conductor $108$
Sign $-0.999 + 0.0424i$
Analytic cond. $6.37220$
Root an. cond. $2.52432$
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  + (1.94 + 2.05i)2-s + (−0.445 + 7.98i)4-s + (−12.1 + 7.04i)5-s + (−21.1 − 12.2i)7-s + (−17.2 + 14.6i)8-s + (−38.1 − 11.3i)10-s + (5.17 − 8.95i)11-s + (22.8 + 39.4i)13-s + (−16.0 − 67.1i)14-s + (−63.6 − 7.11i)16-s − 18.5i·17-s + 142. i·19-s + (−50.8 − 100. i)20-s + (28.4 − 6.78i)22-s + (31.4 + 54.5i)23-s + ⋯
L(s)  = 1  + (0.687 + 0.726i)2-s + (−0.0556 + 0.998i)4-s + (−1.09 + 0.629i)5-s + (−1.14 − 0.659i)7-s + (−0.763 + 0.645i)8-s + (−1.20 − 0.359i)10-s + (0.141 − 0.245i)11-s + (0.486 + 0.842i)13-s + (−0.305 − 1.28i)14-s + (−0.993 − 0.111i)16-s − 0.264i·17-s + 1.72i·19-s + (−0.568 − 1.12i)20-s + (0.275 − 0.0657i)22-s + (0.285 + 0.494i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0424i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.999 + 0.0424i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ -0.999 + 0.0424i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0217541 - 1.02517i\)
\(L(\frac12)\) \(\approx\) \(0.0217541 - 1.02517i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 - 2.05i)T \)
3 \( 1 \)
good5 \( 1 + (12.1 - 7.04i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (21.1 + 12.2i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-5.17 + 8.95i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-22.8 - 39.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 18.5iT - 4.91e3T^{2} \)
19 \( 1 - 142. iT - 6.85e3T^{2} \)
23 \( 1 + (-31.4 - 54.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (28.4 + 16.4i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (72.4 - 41.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 141.T + 5.06e4T^{2} \)
41 \( 1 + (-356. + 205. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-194. - 112. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (190. - 330. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 383. iT - 1.48e5T^{2} \)
59 \( 1 + (-65.2 - 113. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-312. + 541. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-314. + 181. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 280.T + 3.57e5T^{2} \)
73 \( 1 + 178.T + 3.89e5T^{2} \)
79 \( 1 + (595. + 344. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-362. + 627. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 72.7iT - 7.04e5T^{2} \)
97 \( 1 + (434. - 752. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.93399423544869234756037649763, −12.81738640048792835114734204156, −11.86746993411078461338758183050, −10.83117790588702971713918887442, −9.294023665635058299411972328662, −7.85268892894006154242611852741, −7.01790000602785325583664731785, −6.01473784098797821421602497726, −4.05030098719678989131675694067, −3.37234714858705174746858146819, 0.44631247239761172907298284086, 2.86682226762059408745246214263, 4.11461719770153463645176595555, 5.45609782073688518597718002661, 6.82312865045312826565695467909, 8.570128685973434052013917912915, 9.530381158195464124040539213226, 10.86331285080380169465474567584, 11.83789260203176141061775679215, 12.76253238416265027001021213340

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