Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.433 - 0.901i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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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

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.433 - 0.901i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.433 - 0.901i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.03695 + 1.28059i\)
\(L(\frac12)\)  \(\approx\)  \(2.03695 + 1.28059i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 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 \( 1 + (-3.64e3 - 3.06e3i)T + (1.53e7 + 8.71e7i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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