Properties

Label 2-108-108.7-c2-0-8
Degree $2$
Conductor $108$
Sign $-0.962 - 0.269i$
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  + (0.103 + 1.99i)2-s + (0.204 + 2.99i)3-s + (−3.97 + 0.414i)4-s + (1.50 + 8.54i)5-s + (−5.95 + 0.719i)6-s + (7.53 − 8.98i)7-s + (−1.24 − 7.90i)8-s + (−8.91 + 1.22i)9-s + (−16.9 + 3.89i)10-s + (−2.27 − 0.400i)11-s + (−2.05 − 11.8i)12-s + (−4.42 − 1.61i)13-s + (18.7 + 14.1i)14-s + (−25.2 + 6.25i)15-s + (15.6 − 3.30i)16-s + (6.77 + 11.7i)17-s + ⋯
L(s)  = 1  + (0.0519 + 0.998i)2-s + (0.0681 + 0.997i)3-s + (−0.994 + 0.103i)4-s + (0.301 + 1.70i)5-s + (−0.992 + 0.119i)6-s + (1.07 − 1.28i)7-s + (−0.155 − 0.987i)8-s + (−0.990 + 0.136i)9-s + (−1.69 + 0.389i)10-s + (−0.206 − 0.0364i)11-s + (−0.171 − 0.985i)12-s + (−0.340 − 0.123i)13-s + (1.33 + 1.00i)14-s + (−1.68 + 0.417i)15-s + (0.978 − 0.206i)16-s + (0.398 + 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.962 - 0.269i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ -0.962 - 0.269i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.185403 + 1.34888i\)
\(L(\frac12)\) \(\approx\) \(0.185403 + 1.34888i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.103 - 1.99i)T \)
3 \( 1 + (-0.204 - 2.99i)T \)
good5 \( 1 + (-1.50 - 8.54i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-7.53 + 8.98i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (2.27 + 0.400i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (4.42 + 1.61i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-6.77 - 11.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.9 - 8.02i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (3.20 + 3.82i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-34.0 + 12.4i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (0.283 + 0.337i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-18.9 - 32.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-2.35 - 0.858i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (17.5 + 3.10i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-5.64 + 6.72i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 41.4T + 2.80e3T^{2} \)
59 \( 1 + (-25.5 + 4.51i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-51.1 - 42.9i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-32.3 + 88.9i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (100. - 57.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-22.5 + 39.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (19.8 + 54.4i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (22.8 + 62.8i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-54.3 + 94.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-30.2 + 171. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−14.35651130049019766774453793330, −13.61873247853369287727260037366, −11.51109344755795346987874474487, −10.34554723106469807333592180503, −10.03687474956835239971929586347, −8.191763065096214810380661403607, −7.32888867956536924109255529241, −6.05816030898693731973279795830, −4.63185123995302415523007571009, −3.38098767903631055054503859858, 1.13472372025957606102870251351, 2.40330119433818116995004636559, 4.92463601032980749511327185508, 5.52619321784464094312872655634, 7.958050098243106537253281778602, 8.721270409268901915090594678172, 9.491246240086271544982950788184, 11.43879902298206649441700257098, 12.09008931735244068457333650638, 12.67735916252695546940214831989

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