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Properties

Label 2-108-36.23-c1-0-2
Degree $2$
Conductor $108$
Sign $0.995 + 0.0912i$
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  + (1.41 − 0.0786i)2-s + (1.98 − 0.222i)4-s + (−0.686 − 0.396i)5-s + (−2.35 + 1.35i)7-s + (2.78 − 0.469i)8-s + (−0.999 − 0.505i)10-s + (−1.71 − 2.96i)11-s + (−1.68 + 2.92i)13-s + (−3.21 + 2.10i)14-s + (3.90 − 0.882i)16-s + 2.52i·17-s − 2.20i·19-s + (−1.45 − 0.635i)20-s + (−2.65 − 4.05i)22-s + (−1.07 + 1.86i)23-s + ⋯
L(s)  = 1  + (0.998 − 0.0556i)2-s + (0.993 − 0.111i)4-s + (−0.306 − 0.177i)5-s + (−0.888 + 0.513i)7-s + (0.986 − 0.166i)8-s + (−0.316 − 0.159i)10-s + (−0.516 − 0.894i)11-s + (−0.467 + 0.809i)13-s + (−0.858 + 0.561i)14-s + (0.975 − 0.220i)16-s + 0.612i·17-s − 0.506i·19-s + (−0.324 − 0.141i)20-s + (−0.565 − 0.864i)22-s + (−0.224 + 0.388i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.995 + 0.0912i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.995 + 0.0912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58226 - 0.0723411i\)
\(L(\frac12)\) \(\approx\) \(1.58226 - 0.0723411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0786i)T \)
3 \( 1 \)
good5 \( 1 + (0.686 + 0.396i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.35 - 1.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.68 - 2.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.52iT - 17T^{2} \)
19 \( 1 + 2.20iT - 19T^{2} \)
23 \( 1 + (1.07 - 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.686 + 0.396i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.47 - 0.852i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + (0.127 + 0.0737i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.01 + 3.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.77 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 + (-2.58 + 4.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.68 + 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.01 - 3.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.75T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (8.80 - 5.08i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.62 + 6.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.34iT - 89T^{2} \)
97 \( 1 + (-6.24 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.60246670445872334574096527131, −12.70171951713365930002038204197, −11.86478435537130984915150212526, −10.82956201626296192061246438088, −9.524247106825924017860838236826, −8.064221890197588178931808801050, −6.64316346399270093592183553082, −5.63949470376730856046578577406, −4.13811153091525444567144192665, −2.69303974174169208253702718443, 2.80018180209132222668056010980, 4.18782617368989494647911760662, 5.59094306053875691595832194730, 6.97719805364044738143890663501, 7.76520187521465639899563866939, 9.800536183958737503101239053461, 10.63857641326900362625968137880, 11.95975302262222449385326431983, 12.78777531956427191312866083511, 13.58114771342177155039375225696

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