Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.817 - 0.576i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.08 + 3.95i)3-s + (−10.7 − 29.6i)5-s + (15.2 + 86.3i)7-s + (49.7 + 63.9i)9-s + (46.1 − 126. i)11-s + (215. + 180. i)13-s + (29.8 − 282. i)15-s + (300. + 173. i)17-s + (−113. − 196. i)19-s + (−218. + 758. i)21-s + (−187. − 33.0i)23-s + (−283. + 237. i)25-s + (149. + 713. i)27-s + (429. + 512. i)29-s + (−35.8 + 203. i)31-s + ⋯
L(s)  = 1  + (0.898 + 0.439i)3-s + (−0.431 − 1.18i)5-s + (0.310 + 1.76i)7-s + (0.614 + 0.788i)9-s + (0.381 − 1.04i)11-s + (1.27 + 1.07i)13-s + (0.132 − 1.25i)15-s + (1.03 + 0.599i)17-s + (−0.315 − 0.545i)19-s + (−0.494 + 1.72i)21-s + (−0.354 − 0.0625i)23-s + (−0.453 + 0.380i)25-s + (0.205 + 0.978i)27-s + (0.511 + 0.609i)29-s + (−0.0373 + 0.211i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.817 - 0.576i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.817 - 0.576i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.28924 + 0.726320i\)
\(L(\frac12)\)  \(\approx\)  \(2.28924 + 0.726320i\)
\(L(3)\)   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 \( 1 + (-8.08 - 3.95i)T \)
good5 \( 1 + (10.7 + 29.6i)T + (-478. + 401. i)T^{2} \)
7 \( 1 + (-15.2 - 86.3i)T + (-2.25e3 + 821. i)T^{2} \)
11 \( 1 + (-46.1 + 126. i)T + (-1.12e4 - 9.41e3i)T^{2} \)
13 \( 1 + (-215. - 180. i)T + (4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (-300. - 173. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (113. + 196. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (187. + 33.0i)T + (2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (-429. - 512. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (35.8 - 203. i)T + (-8.67e5 - 3.15e5i)T^{2} \)
37 \( 1 + (-987. + 1.71e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (1.25e3 - 1.49e3i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (793. + 288. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (1.80e3 - 317. i)T + (4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 + 2.96e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.88e3 + 5.18e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (-680. - 3.86e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (3.95e3 + 3.32e3i)T + (3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (1.16e3 + 671. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (1.15e3 + 1.99e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-6.08e3 + 5.10e3i)T + (6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (879. + 1.04e3i)T + (-8.24e6 + 4.67e7i)T^{2} \)
89 \( 1 + (2.80e3 - 1.61e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (1.03e4 + 3.75e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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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

−13.07605925495950534908553311744, −12.09615291549690611915893428314, −11.15053394449749224718765878307, −9.383565204854699400062375036212, −8.591649661575585832601968270788, −8.326079833320509845417669880323, −6.05765659961991742750218410050, −4.79016456602446725172897759165, −3.41976489014673401023438211126, −1.63194605453769403590889744908, 1.19063678742368227188042681770, 3.19314398917504812840148563734, 4.09836742547992218687664021631, 6.54173324177923981036138436840, 7.46199273341408055055306972604, 8.071108856727521320487317805047, 9.950252114452534556472235074356, 10.54399963625298619892188490354, 11.83228392345341793136974021677, 13.19554578840733743395016581481

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