Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.590 + 0.807i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.1 + 22.7i)5-s + (−31.6 − 54.7i)7-s + (49.1 + 85.0i)11-s + (369. − 639. i)13-s − 250.·17-s + 1.10e3·19-s + (2.20e3 − 3.81e3i)23-s + (1.21e3 + 2.10e3i)25-s + (−3.94e3 − 6.82e3i)29-s + (2.30e3 − 3.99e3i)31-s + 1.66e3·35-s + 1.18e4·37-s + (5.04e3 − 8.73e3i)41-s + (−3.51e3 − 6.09e3i)43-s + (7.45e3 + 1.29e4i)47-s + ⋯
L(s)  = 1  + (−0.235 + 0.407i)5-s + (−0.243 − 0.422i)7-s + (0.122 + 0.211i)11-s + (0.605 − 1.04i)13-s − 0.209·17-s + 0.700·19-s + (0.869 − 1.50i)23-s + (0.389 + 0.674i)25-s + (−0.870 − 1.50i)29-s + (0.430 − 0.746i)31-s + 0.229·35-s + 1.42·37-s + (0.468 − 0.811i)41-s + (−0.290 − 0.502i)43-s + (0.492 + 0.853i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.590 + 0.807i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.590 + 0.807i)\)
\(L(3)\)  \(\approx\)  \(1.42570 - 0.723620i\)
\(L(\frac12)\)  \(\approx\)  \(1.42570 - 0.723620i\)
\(L(\frac{7}{2})\)   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 \( 1 \)
good5 \( 1 + (13.1 - 22.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (31.6 + 54.7i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-49.1 - 85.0i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-369. + 639. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 250.T + 1.41e6T^{2} \)
19 \( 1 - 1.10e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.20e3 + 3.81e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (3.94e3 + 6.82e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-2.30e3 + 3.99e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.18e4T + 6.93e7T^{2} \)
41 \( 1 + (-5.04e3 + 8.73e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (3.51e3 + 6.09e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-7.45e3 - 1.29e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 2.24e4T + 4.18e8T^{2} \)
59 \( 1 + (-5.40e3 + 9.36e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-594. - 1.02e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.95e4 - 5.12e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + 5.30e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.86e4 - 3.23e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-6.04e4 - 1.04e5i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 9.78e4T + 5.58e9T^{2} \)
97 \( 1 + (5.33e4 + 9.24e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.74594218631121509444606874825, −11.40035792735846070903463617903, −10.57790847512600903730317441094, −9.459461454650142027603209416041, −8.098965247408040045874069200854, −7.05284214421958046537630674461, −5.79294652754059386781440475868, −4.17402897520050080214596573639, −2.80808368853471855466939628982, −0.69289580725881483841192344883, 1.33096837679867960779078959382, 3.24600332897800041844240350576, 4.73380500625216229512864829187, 6.08822816563036391548888912487, 7.36596983937826516821425933870, 8.763879628615398619936395951737, 9.448753887713059718437711033928, 11.00569532344850065761628624756, 11.83054458038146397895878383059, 12.90958505647163949182349455080

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