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Properties

Label 2-108-9.4-c3-0-0
Degree $2$
Conductor $108$
Sign $0.241 - 0.970i$
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  + (−2.44 + 4.24i)5-s + (5.32 + 9.22i)7-s + (17.7 + 30.7i)11-s + (−36.3 + 62.9i)13-s + 127.·17-s − 46.3·19-s + (−65.5 + 113. i)23-s + (50.5 + 87.4i)25-s + (−68.7 − 119. i)29-s + (53.0 − 91.9i)31-s − 52.1·35-s + 137.·37-s + (−35.8 + 62.1i)41-s + (−188. − 326. i)43-s + (−306. − 531. i)47-s + ⋯
L(s)  = 1  + (−0.219 + 0.379i)5-s + (0.287 + 0.498i)7-s + (0.485 + 0.841i)11-s + (−0.775 + 1.34i)13-s + 1.81·17-s − 0.560·19-s + (−0.594 + 1.02i)23-s + (0.404 + 0.699i)25-s + (−0.440 − 0.762i)29-s + (0.307 − 0.532i)31-s − 0.252·35-s + 0.610·37-s + (−0.136 + 0.236i)41-s + (−0.668 − 1.15i)43-s + (−0.952 − 1.64i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.12518 + 0.879927i\)
\(L(\frac12)\) \(\approx\) \(1.12518 + 0.879927i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (2.44 - 4.24i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-5.32 - 9.22i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-17.7 - 30.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (36.3 - 62.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 127.T + 4.91e3T^{2} \)
19 \( 1 + 46.3T + 6.85e3T^{2} \)
23 \( 1 + (65.5 - 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (68.7 + 119. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-53.0 + 91.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 137.T + 5.06e4T^{2} \)
41 \( 1 + (35.8 - 62.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (188. + 326. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (306. + 531. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 431.T + 1.48e5T^{2} \)
59 \( 1 + (142. - 247. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-21.9 - 38.0i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-22.6 + 39.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 357.T + 3.57e5T^{2} \)
73 \( 1 - 530.T + 3.89e5T^{2} \)
79 \( 1 + (-97.5 - 168. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-380. - 658. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.21e3T + 7.04e5T^{2} \)
97 \( 1 + (-552. - 956. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.51964286725274008310458595711, −12.00218071368992990265877012755, −11.74386124722269177645984727946, −10.11298442875650763935324691896, −9.285469010819274202718184661817, −7.83042344170278209602709142547, −6.82786670829519917594680307417, −5.33381873979545944384191600159, −3.86032498225292129408747640846, −1.98419366211113976054550999618, 0.838075709161267440041902480326, 3.19849824177341612019070204320, 4.75352069789129608425317439024, 6.07135521324305211373277146049, 7.66364455059587464179483954627, 8.442024772881548228542011118635, 9.933622607779295860126863290506, 10.80206304797634604973418199055, 12.12785534951389065258410022600, 12.82502953211679163752117895769

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