Properties

Label 2-108-27.11-c4-0-9
Degree $2$
Conductor $108$
Sign $0.316 + 0.948i$
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  + (8.73 − 2.16i)3-s + (9.54 − 26.2i)5-s + (−0.541 + 3.07i)7-s + (71.6 − 37.8i)9-s + (−44.3 − 121. i)11-s + (−30.3 + 25.5i)13-s + (26.5 − 249. i)15-s + (−72.0 + 41.5i)17-s + (32.6 − 56.5i)19-s + (1.92 + 28.0i)21-s + (346. − 61.0i)23-s + (−117. − 98.5i)25-s + (543. − 485. i)27-s + (421. − 501. i)29-s + (−18.7 − 106. i)31-s + ⋯
L(s)  = 1  + (0.970 − 0.240i)3-s + (0.381 − 1.04i)5-s + (−0.0110 + 0.0627i)7-s + (0.884 − 0.467i)9-s + (−0.366 − 1.00i)11-s + (−0.179 + 0.150i)13-s + (0.118 − 1.10i)15-s + (−0.249 + 0.143i)17-s + (0.0904 − 0.156i)19-s + (0.00436 + 0.0635i)21-s + (0.654 − 0.115i)23-s + (−0.187 − 0.157i)25-s + (0.745 − 0.666i)27-s + (0.500 − 0.596i)29-s + (−0.0194 − 0.110i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(11.1639\)
Root analytic conductor: \(3.34125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :2),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.99804 - 1.43985i\)
\(L(\frac12)\) \(\approx\) \(1.99804 - 1.43985i\)
\(L(3)\) 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 \( 1 + (-8.73 + 2.16i)T \)
good5 \( 1 + (-9.54 + 26.2i)T + (-478. - 401. i)T^{2} \)
7 \( 1 + (0.541 - 3.07i)T + (-2.25e3 - 821. i)T^{2} \)
11 \( 1 + (44.3 + 121. i)T + (-1.12e4 + 9.41e3i)T^{2} \)
13 \( 1 + (30.3 - 25.5i)T + (4.95e3 - 2.81e4i)T^{2} \)
17 \( 1 + (72.0 - 41.5i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-32.6 + 56.5i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-346. + 61.0i)T + (2.62e5 - 9.57e4i)T^{2} \)
29 \( 1 + (-421. + 501. i)T + (-1.22e5 - 6.96e5i)T^{2} \)
31 \( 1 + (18.7 + 106. i)T + (-8.67e5 + 3.15e5i)T^{2} \)
37 \( 1 + (745. + 1.29e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-1.34e3 - 1.59e3i)T + (-4.90e5 + 2.78e6i)T^{2} \)
43 \( 1 + (1.40e3 - 511. i)T + (2.61e6 - 2.19e6i)T^{2} \)
47 \( 1 + (-2.36e3 - 416. i)T + (4.58e6 + 1.66e6i)T^{2} \)
53 \( 1 - 2.34e3iT - 7.89e6T^{2} \)
59 \( 1 + (978. - 2.68e3i)T + (-9.28e6 - 7.78e6i)T^{2} \)
61 \( 1 + (1.13e3 - 6.46e3i)T + (-1.30e7 - 4.73e6i)T^{2} \)
67 \( 1 + (6.61e3 - 5.54e3i)T + (3.49e6 - 1.98e7i)T^{2} \)
71 \( 1 + (426. - 245. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (2.00e3 - 3.47e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (5.37e3 + 4.50e3i)T + (6.76e6 + 3.83e7i)T^{2} \)
83 \( 1 + (-1.60e3 + 1.91e3i)T + (-8.24e6 - 4.67e7i)T^{2} \)
89 \( 1 + (-1.29e4 - 7.46e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.58e4 + 5.77e3i)T + (6.78e7 - 5.69e7i)T^{2} \)
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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

−13.08521035995730531410212713064, −11.99408568974110036959862076906, −10.49533836095236063859289711709, −9.159679523539565737006897477494, −8.646052986449283124620362212193, −7.43433294315008576125272045437, −5.85624791024350711161473459975, −4.39332199220652339478642596808, −2.73994271927428490758040759082, −1.07027426412423050308933742338, 2.11490853659349884716606318102, 3.27995509445915042039511895455, 4.87609075809365941701091317990, 6.74010340063684153994560463904, 7.59392616774786879636885033594, 8.968314257035727580660223407518, 10.07087169719432881220770637781, 10.70007988692356422512732539056, 12.31394272609731345808689209340, 13.43306744511473562199000443541

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