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Properties

Label 2-108-27.16-c1-0-2
Degree $2$
Conductor $108$
Sign $-0.584 + 0.811i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43 − 0.972i)3-s + (−3.94 − 1.43i)5-s + (0.610 − 3.46i)7-s + (1.10 + 2.78i)9-s + (1.73 − 0.632i)11-s + (−1.78 + 1.49i)13-s + (4.26 + 5.89i)15-s + (−0.799 − 1.38i)17-s + (2.31 − 4.00i)19-s + (−4.24 + 4.36i)21-s + (−0.308 − 1.74i)23-s + (9.68 + 8.12i)25-s + (1.12 − 5.07i)27-s + (0.882 + 0.740i)29-s + (0.322 + 1.83i)31-s + ⋯
L(s)  = 1  + (−0.827 − 0.561i)3-s + (−1.76 − 0.642i)5-s + (0.230 − 1.30i)7-s + (0.369 + 0.929i)9-s + (0.523 − 0.190i)11-s + (−0.495 + 0.415i)13-s + (1.10 + 1.52i)15-s + (−0.194 − 0.336i)17-s + (0.530 − 0.918i)19-s + (−0.925 + 0.953i)21-s + (−0.0642 − 0.364i)23-s + (1.93 + 1.62i)25-s + (0.215 − 0.976i)27-s + (0.163 + 0.137i)29-s + (0.0579 + 0.328i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.584 + 0.811i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ -0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238258 - 0.465528i\)
\(L(\frac12)\) \(\approx\) \(0.238258 - 0.465528i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 0.972i)T \)
good5 \( 1 + (3.94 + 1.43i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.610 + 3.46i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.73 + 0.632i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (1.78 - 1.49i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.799 + 1.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.31 + 4.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.308 + 1.74i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.882 - 0.740i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.322 - 1.83i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.38 + 7.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.98 + 2.50i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.41 - 0.880i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.29 + 7.35i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 8.02T + 53T^{2} \)
59 \( 1 + (1.15 + 0.418i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.754 - 4.28i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (4.86 - 4.08i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.871 - 1.50i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.63 + 6.40i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.65 - 7.25i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (2.71 - 4.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.3 + 4.13i)T + (74.3 - 62.3i)T^{2} \)
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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

−13.15204857920569784529910150435, −12.07008360541736934199308852480, −11.48783896914276969055717597500, −10.57574658583325899549706992635, −8.772104442918070866435722708658, −7.46048090754047967899816760322, −7.02760104877031970902768549354, −4.92720351933321895827445232407, −3.99818727954484975017416120374, −0.68124335646718234286427935157, 3.36858322522680001921198107370, 4.67143517459750732666823949471, 6.08760128558349206448272934302, 7.44541825683599381368651188769, 8.614149475680777829485927998019, 10.01763713401807479590618433736, 11.21729801176347981210323569016, 11.91635564791021720790992738473, 12.38823428099632675497112972379, 14.58445605474397982106461150304

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