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Properties

Label 2-108-36.23-c3-0-14
Degree $2$
Conductor $108$
Sign $0.361 + 0.932i$
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  + (2.75 − 0.655i)2-s + (7.14 − 3.60i)4-s + (−12.1 − 7.04i)5-s + (21.1 − 12.2i)7-s + (17.2 − 14.6i)8-s + (−38.1 − 11.3i)10-s + (−5.17 − 8.95i)11-s + (22.8 − 39.4i)13-s + (50.1 − 47.4i)14-s + (37.9 − 51.5i)16-s + 18.5i·17-s + 142. i·19-s + (−112. − 6.26i)20-s + (−20.1 − 21.2i)22-s + (−31.4 + 54.5i)23-s + ⋯
L(s)  = 1  + (0.972 − 0.231i)2-s + (0.892 − 0.451i)4-s + (−1.09 − 0.629i)5-s + (1.14 − 0.659i)7-s + (0.763 − 0.645i)8-s + (−1.20 − 0.359i)10-s + (−0.141 − 0.245i)11-s + (0.486 − 0.842i)13-s + (0.958 − 0.906i)14-s + (0.593 − 0.805i)16-s + 0.264i·17-s + 1.72i·19-s + (−1.25 − 0.0700i)20-s + (−0.194 − 0.205i)22-s + (−0.285 + 0.494i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.16708 - 1.48411i\)
\(L(\frac12)\) \(\approx\) \(2.16708 - 1.48411i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.75 + 0.655i)T \)
3 \( 1 \)
good5 \( 1 + (12.1 + 7.04i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-21.1 + 12.2i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (5.17 + 8.95i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.8 + 39.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 18.5iT - 4.91e3T^{2} \)
19 \( 1 - 142. iT - 6.85e3T^{2} \)
23 \( 1 + (31.4 - 54.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (28.4 - 16.4i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-72.4 - 41.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 141.T + 5.06e4T^{2} \)
41 \( 1 + (-356. - 205. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (194. - 112. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-190. - 330. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 383. iT - 1.48e5T^{2} \)
59 \( 1 + (65.2 - 113. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-312. - 541. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (314. + 181. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 280.T + 3.57e5T^{2} \)
73 \( 1 + 178.T + 3.89e5T^{2} \)
79 \( 1 + (-595. + 344. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (362. + 627. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 72.7iT - 7.04e5T^{2} \)
97 \( 1 + (434. + 752. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.93975233481083907484606125727, −12.03203895307957367187650406265, −11.19259678648052995375916672372, −10.28458310060069533148438548565, −8.208256701031090702498599826563, −7.62832709074011413040388617516, −5.81141829941399769792270370335, −4.55957215022096066014566692184, −3.60464814143949344422112682333, −1.26502902014118699134490922267, 2.40366776770639047869090229999, 4.02497424857425414499156802082, 5.10153520828583412107710692829, 6.70263754020349113224261792957, 7.65325914125014746893098520091, 8.760558481735108496796355659272, 10.89618207129156596315923620635, 11.47660372499650203023337393927, 12.20495453433739733800558430429, 13.61030809715872140894758351468

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