Properties

Label 2-108-27.2-c4-0-11
Degree $2$
Conductor $108$
Sign $-0.999 - 0.00124i$
Analytic cond. $11.1639$
Root an. cond. $3.34125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.21 − 8.91i)3-s + (−22.3 − 26.6i)5-s + (80.4 − 29.2i)7-s + (−78.0 + 21.6i)9-s + (−32.7 + 39.0i)11-s + (−24.9 − 141. i)13-s + (−210. + 231. i)15-s + (−462. + 266. i)17-s + (16.9 − 29.4i)19-s + (−359. − 682. i)21-s + (7.86 − 21.6i)23-s + (−101. + 575. i)25-s + (287. + 669. i)27-s + (890. + 157. i)29-s + (−1.36e3 − 495. i)31-s + ⋯
L(s)  = 1  + (−0.134 − 0.990i)3-s + (−0.894 − 1.06i)5-s + (1.64 − 0.597i)7-s + (−0.963 + 0.267i)9-s + (−0.270 + 0.322i)11-s + (−0.147 − 0.838i)13-s + (−0.935 + 1.02i)15-s + (−1.59 + 0.923i)17-s + (0.0470 − 0.0815i)19-s + (−0.814 − 1.54i)21-s + (0.0148 − 0.0408i)23-s + (−0.162 + 0.921i)25-s + (0.394 + 0.918i)27-s + (1.05 + 0.186i)29-s + (−1.41 − 0.516i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00124i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.999 - 0.00124i$
Analytic conductor: \(11.1639\)
Root analytic conductor: \(3.34125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :2),\ -0.999 - 0.00124i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.000640981 + 1.02585i\)
\(L(\frac12)\) \(\approx\) \(0.000640981 + 1.02585i\)
\(L(3)\) 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 \( 1 + (1.21 + 8.91i)T \)
good5 \( 1 + (22.3 + 26.6i)T + (-108. + 615. i)T^{2} \)
7 \( 1 + (-80.4 + 29.2i)T + (1.83e3 - 1.54e3i)T^{2} \)
11 \( 1 + (32.7 - 39.0i)T + (-2.54e3 - 1.44e4i)T^{2} \)
13 \( 1 + (24.9 + 141. i)T + (-2.68e4 + 9.76e3i)T^{2} \)
17 \( 1 + (462. - 266. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-16.9 + 29.4i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-7.86 + 21.6i)T + (-2.14e5 - 1.79e5i)T^{2} \)
29 \( 1 + (-890. - 157. i)T + (6.64e5 + 2.41e5i)T^{2} \)
31 \( 1 + (1.36e3 + 495. i)T + (7.07e5 + 5.93e5i)T^{2} \)
37 \( 1 + (500. + 866. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (316. - 55.7i)T + (2.65e6 - 9.66e5i)T^{2} \)
43 \( 1 + (-1.51e3 - 1.26e3i)T + (5.93e5 + 3.36e6i)T^{2} \)
47 \( 1 + (1.09e3 + 3.01e3i)T + (-3.73e6 + 3.13e6i)T^{2} \)
53 \( 1 + 256. iT - 7.89e6T^{2} \)
59 \( 1 + (-470. - 560. i)T + (-2.10e6 + 1.19e7i)T^{2} \)
61 \( 1 + (-6.42e3 + 2.33e3i)T + (1.06e7 - 8.89e6i)T^{2} \)
67 \( 1 + (37.1 + 210. i)T + (-1.89e7 + 6.89e6i)T^{2} \)
71 \( 1 + (-1.97e3 + 1.13e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (2.32e3 - 4.02e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-1.83e3 + 1.03e4i)T + (-3.66e7 - 1.33e7i)T^{2} \)
83 \( 1 + (2.37e3 + 419. i)T + (4.45e7 + 1.62e7i)T^{2} \)
89 \( 1 + (1.22e3 + 706. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (9.76e3 + 8.19e3i)T + (1.53e7 + 8.71e7i)T^{2} \)
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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

−12.55186267791185363484083919719, −11.49417322468694994972613435314, −10.76734702132585010284858316321, −8.598392832424760985867021539179, −8.110558201076055720778395072873, −7.14357882805045165156465668545, −5.31504865257897940708180763402, −4.28896298425378196155640923982, −1.83727129204222592845121578403, −0.46247227214617546231846706066, 2.55439203576240125881829363209, 4.15689939693812718883863364944, 5.16414549597900224616101281667, 6.86385823253753052567598154651, 8.186343402532224548015482329643, 9.129889585906021098240032105521, 10.75435918645501618311487551815, 11.26637012171141014567898904995, 11.88688703657837043757623778839, 13.99895068599686122361284056731

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