Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.302 + 0.953i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 5.19i)3-s + (6.54 − 5.48i)5-s + (11.9 − 4.35i)7-s + (−26.9 + 1.79i)9-s + (−7.56 − 6.34i)11-s + (−6.10 − 34.6i)13-s + (−29.6 − 33.0i)15-s + (−44.7 − 77.4i)17-s + (−1.00 + 1.73i)19-s + (−24.6 − 61.3i)21-s + (33.9 + 12.3i)23-s + (−9.04 + 51.3i)25-s + (13.9 + 139. i)27-s + (40.5 − 230. i)29-s + (113. + 41.1i)31-s + ⋯
L(s)  = 1  + (−0.0333 − 0.999i)3-s + (0.585 − 0.490i)5-s + (0.646 − 0.235i)7-s + (−0.997 + 0.0665i)9-s + (−0.207 − 0.173i)11-s + (−0.130 − 0.738i)13-s + (−0.510 − 0.568i)15-s + (−0.637 − 1.10i)17-s + (−0.0121 + 0.0210i)19-s + (−0.256 − 0.637i)21-s + (0.308 + 0.112i)23-s + (−0.0723 + 0.410i)25-s + (0.0997 + 0.995i)27-s + (0.259 − 1.47i)29-s + (0.655 + 0.238i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.302 + 0.953i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.302 + 0.953i)\)
\(L(2)\)  \(\approx\)  \(0.947311 - 1.29453i\)
\(L(\frac12)\)  \(\approx\)  \(0.947311 - 1.29453i\)
\(L(\frac{5}{2})\)   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 \( 1 + (0.173 + 5.19i)T \)
good5 \( 1 + (-6.54 + 5.48i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-11.9 + 4.35i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (7.56 + 6.34i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (6.10 + 34.6i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (44.7 + 77.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (1.00 - 1.73i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-33.9 - 12.3i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (-40.5 + 230. i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-113. - 41.1i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (-98.7 - 170. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (4.40 + 25.0i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (-410. - 344. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (17.7 - 6.44i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 - 366.T + 1.48e5T^{2} \)
59 \( 1 + (122. - 102. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (-385. + 140. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (-152. - 864. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (-69.1 - 119. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-554. + 959. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (48.8 - 276. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-168. + 958. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (-530. + 919. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (158. + 133. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−13.14715391614837655410962915481, −11.93037386494232846337520746162, −11.02575759235794188488228361061, −9.573876780322873035417656794225, −8.355214577122052569159710749689, −7.42119814339140148151682273903, −6.05018545825276246686836958172, −4.89428644910370636861231149162, −2.56583933066714859361748425337, −0.932001144327614861707209639759, 2.32405397787771880520850342995, 4.10471673610282119490895178543, 5.35709708418722174168277214401, 6.62368634540876821592443675997, 8.339442173347833587384591935602, 9.324996618067552082459338168037, 10.46416602460810021897548951910, 11.11519294451990104049727716743, 12.38018906266001909693811429359, 13.85896539862644825891336335960

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