Properties

Label 2-108-36.11-c3-0-3
Degree $2$
Conductor $108$
Sign $0.292 - 0.956i$
Analytic cond. $6.37220$
Root an. cond. $2.52432$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 + 0.934i)2-s + (6.25 − 4.98i)4-s + (−4.30 + 2.48i)5-s + (3.07 + 1.77i)7-s + (−12.0 + 19.1i)8-s + (9.16 − 10.6i)10-s + (22.1 − 38.4i)11-s + (30.6 + 53.1i)13-s + (−9.85 − 1.86i)14-s + (14.2 − 62.4i)16-s + 99.9i·17-s + 85.6i·19-s + (−14.5 + 37.0i)20-s + (−23.3 + 123. i)22-s + (41.3 + 71.5i)23-s + ⋯
L(s)  = 1  + (−0.943 + 0.330i)2-s + (0.781 − 0.623i)4-s + (−0.385 + 0.222i)5-s + (0.165 + 0.0957i)7-s + (−0.531 + 0.846i)8-s + (0.289 − 0.337i)10-s + (0.608 − 1.05i)11-s + (0.654 + 1.13i)13-s + (−0.188 − 0.0355i)14-s + (0.221 − 0.975i)16-s + 1.42i·17-s + 1.03i·19-s + (−0.162 + 0.413i)20-s + (−0.225 + 1.19i)22-s + (0.374 + 0.648i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.761949 + 0.563501i\)
\(L(\frac12)\) \(\approx\) \(0.761949 + 0.563501i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.66 - 0.934i)T \)
3 \( 1 \)
good5 \( 1 + (4.30 - 2.48i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-3.07 - 1.77i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-22.1 + 38.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-30.6 - 53.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 99.9iT - 4.91e3T^{2} \)
19 \( 1 - 85.6iT - 6.85e3T^{2} \)
23 \( 1 + (-41.3 - 71.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-152. - 88.3i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-171. + 98.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 172.T + 5.06e4T^{2} \)
41 \( 1 + (38.2 - 22.0i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-67.7 - 39.0i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-229. + 398. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 290. iT - 1.48e5T^{2} \)
59 \( 1 + (-147. - 256. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (247. - 428. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (19.4 - 11.2i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 304.T + 3.57e5T^{2} \)
73 \( 1 - 1.16e3T + 3.89e5T^{2} \)
79 \( 1 + (964. + 556. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (400. - 694. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 346. iT - 7.04e5T^{2} \)
97 \( 1 + (-291. + 504. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.68675547725637599043305734072, −11.94699091005385585253075002921, −11.24439690808364262376765329016, −10.21054231676006161563863743448, −8.870509924618512336141165443493, −8.190683148730734262245054978225, −6.78605178297761910634269328199, −5.82175710555337806079667516309, −3.68691826215852240460298259592, −1.48792392796386617369885102875, 0.793071946469265362648472798855, 2.79848763356262619442110851479, 4.57528212620374830262187226411, 6.58542806594224571662680867251, 7.66376021234219150435369440528, 8.717851943238992231482822395064, 9.769816059025500951294272305286, 10.81620796927825516629631673887, 11.86746822169044597539923395064, 12.62957865239235405884977723741

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