Properties

Label 2-108-27.5-c2-0-0
Degree $2$
Conductor $108$
Sign $0.252 - 0.967i$
Analytic cond. $2.94278$
Root an. cond. $1.71545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.92 − 0.656i)3-s + (0.740 + 2.03i)5-s + (1.08 + 6.13i)7-s + (8.13 + 3.84i)9-s + (−5.10 + 14.0i)11-s + (9.95 + 8.35i)13-s + (−0.831 − 6.44i)15-s + (−3.36 − 1.94i)17-s + (6.39 + 11.0i)19-s + (0.863 − 18.6i)21-s + (−35.3 − 6.23i)23-s + (15.5 − 13.0i)25-s + (−21.2 − 16.6i)27-s + (7.13 + 8.49i)29-s + (6.25 − 35.4i)31-s + ⋯
L(s)  = 1  + (−0.975 − 0.218i)3-s + (0.148 + 0.407i)5-s + (0.154 + 0.877i)7-s + (0.904 + 0.427i)9-s + (−0.464 + 1.27i)11-s + (0.766 + 0.642i)13-s + (−0.0554 − 0.429i)15-s + (−0.197 − 0.114i)17-s + (0.336 + 0.583i)19-s + (0.0411 − 0.889i)21-s + (−1.53 − 0.271i)23-s + (0.622 − 0.522i)25-s + (−0.788 − 0.614i)27-s + (0.245 + 0.293i)29-s + (0.201 − 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.738790 + 0.570892i\)
\(L(\frac12)\) \(\approx\) \(0.738790 + 0.570892i\)
\(L(2)\) 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 \( 1 + (2.92 + 0.656i)T \)
good5 \( 1 + (-0.740 - 2.03i)T + (-19.1 + 16.0i)T^{2} \)
7 \( 1 + (-1.08 - 6.13i)T + (-46.0 + 16.7i)T^{2} \)
11 \( 1 + (5.10 - 14.0i)T + (-92.6 - 77.7i)T^{2} \)
13 \( 1 + (-9.95 - 8.35i)T + (29.3 + 166. i)T^{2} \)
17 \( 1 + (3.36 + 1.94i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.39 - 11.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (35.3 + 6.23i)T + (497. + 180. i)T^{2} \)
29 \( 1 + (-7.13 - 8.49i)T + (-146. + 828. i)T^{2} \)
31 \( 1 + (-6.25 + 35.4i)T + (-903. - 328. i)T^{2} \)
37 \( 1 + (16.3 - 28.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (14.5 - 17.3i)T + (-291. - 1.65e3i)T^{2} \)
43 \( 1 + (-58.2 - 21.2i)T + (1.41e3 + 1.18e3i)T^{2} \)
47 \( 1 + (3.36 - 0.592i)T + (2.07e3 - 755. i)T^{2} \)
53 \( 1 + 79.3iT - 2.80e3T^{2} \)
59 \( 1 + (-7.36 - 20.2i)T + (-2.66e3 + 2.23e3i)T^{2} \)
61 \( 1 + (-13.7 - 77.7i)T + (-3.49e3 + 1.27e3i)T^{2} \)
67 \( 1 + (84.5 + 70.9i)T + (779. + 4.42e3i)T^{2} \)
71 \( 1 + (-71.2 - 41.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (3.09 + 5.36i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-49.2 + 41.3i)T + (1.08e3 - 6.14e3i)T^{2} \)
83 \( 1 + (-97.6 - 116. i)T + (-1.19e3 + 6.78e3i)T^{2} \)
89 \( 1 + (-97.5 + 56.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (52.0 + 18.9i)T + (7.20e3 + 6.04e3i)T^{2} \)
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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

−13.54304733112077027764754825392, −12.36301687870438558283784595319, −11.76718700903659784317414843484, −10.57674735612975540033698530857, −9.648354151810621401453523998081, −8.073950690217211056858875574411, −6.75830145337255066409488796330, −5.78169410846402643323307518877, −4.44732023113174156399353744956, −2.06816420905515874500524229693, 0.813815529805682718239646765904, 3.73738118984307203280689899575, 5.19601465855950358137024351748, 6.21088227212213106709898898365, 7.64462450985235843230506651849, 8.963085765104479786785902231871, 10.47569759454770921903079391882, 10.90828678692164474723102064879, 12.13933244881135796289351917696, 13.26363364093008458845354403808

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