Properties

Degree $2$
Conductor $108$
Sign $0.433 + 0.901i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (4.39 − 7.85i)3-s + (6.32 − 7.53i)5-s + (66.1 + 24.0i)7-s + (−42.4 − 69.0i)9-s + (90.1 + 107. i)11-s + (37.0 − 209. i)13-s + (−31.4 − 82.7i)15-s + (−45.6 − 26.3i)17-s + (−163. − 283. i)19-s + (479. − 413. i)21-s + (−193. − 530. i)23-s + (91.7 + 520. i)25-s + (−728. + 29.9i)27-s + (109. − 19.3i)29-s + (868. − 316. i)31-s + ⋯
L(s)  = 1  + (0.488 − 0.872i)3-s + (0.252 − 0.301i)5-s + (1.34 + 0.491i)7-s + (−0.523 − 0.852i)9-s + (0.745 + 0.887i)11-s + (0.218 − 1.24i)13-s + (−0.139 − 0.367i)15-s + (−0.157 − 0.0912i)17-s + (−0.453 − 0.784i)19-s + (1.08 − 0.937i)21-s + (−0.365 − 1.00i)23-s + (0.146 + 0.832i)25-s + (−0.999 + 0.0410i)27-s + (0.130 − 0.0230i)29-s + (0.903 − 0.329i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.433 + 0.901i$
Motivic weight: \(4\)
Character: $\chi_{108} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :2),\ 0.433 + 0.901i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.03695 - 1.28059i\)
\(L(\frac12)\) \(\approx\) \(2.03695 - 1.28059i\)
\(L(3)\) 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 \( 1 + (-4.39 + 7.85i)T \)
good5 \( 1 + (-6.32 + 7.53i)T + (-108. - 615. i)T^{2} \)
7 \( 1 + (-66.1 - 24.0i)T + (1.83e3 + 1.54e3i)T^{2} \)
11 \( 1 + (-90.1 - 107. i)T + (-2.54e3 + 1.44e4i)T^{2} \)
13 \( 1 + (-37.0 + 209. i)T + (-2.68e4 - 9.76e3i)T^{2} \)
17 \( 1 + (45.6 + 26.3i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (163. + 283. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (193. + 530. i)T + (-2.14e5 + 1.79e5i)T^{2} \)
29 \( 1 + (-109. + 19.3i)T + (6.64e5 - 2.41e5i)T^{2} \)
31 \( 1 + (-868. + 316. i)T + (7.07e5 - 5.93e5i)T^{2} \)
37 \( 1 + (-154. + 267. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (478. + 84.2i)T + (2.65e6 + 9.66e5i)T^{2} \)
43 \( 1 + (-1.94e3 + 1.63e3i)T + (5.93e5 - 3.36e6i)T^{2} \)
47 \( 1 + (1.39e3 - 3.82e3i)T + (-3.73e6 - 3.13e6i)T^{2} \)
53 \( 1 - 4.27e3iT - 7.89e6T^{2} \)
59 \( 1 + (3.17e3 - 3.78e3i)T + (-2.10e6 - 1.19e7i)T^{2} \)
61 \( 1 + (1.47e3 + 536. i)T + (1.06e7 + 8.89e6i)T^{2} \)
67 \( 1 + (527. - 2.98e3i)T + (-1.89e7 - 6.89e6i)T^{2} \)
71 \( 1 + (5.04e3 + 2.91e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-3.49e3 - 6.05e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-54.1 - 307. i)T + (-3.66e7 + 1.33e7i)T^{2} \)
83 \( 1 + (-1.18e4 + 2.08e3i)T + (4.45e7 - 1.62e7i)T^{2} \)
89 \( 1 + (8.39e3 - 4.84e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-3.64e3 + 3.06e3i)T + (1.53e7 - 8.71e7i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.73873737996344688251919410314, −12.00653049949382954865313286935, −10.84421714169782247264552871032, −9.244132712237945793575624134616, −8.372130807110392028475220245145, −7.40663739314529389259405199803, −6.00347200545529676963413403481, −4.58498382646755524002662047966, −2.50437661783754926129639401332, −1.20219268084321148814073075547, 1.78690001902177954512024737521, 3.70512146931850935649621463493, 4.74416631117763010771687938268, 6.31763654235285104781639503862, 7.970087100048883992918933847811, 8.806447610870827956035060299175, 10.01594659764058747205027816581, 11.05835108802100939143349888259, 11.73055401481078316454940146054, 13.77687848394952969643474833619

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