Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.620 + 0.783i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.88 + 2.10i)2-s + (2.88 − 4.31i)3-s + (−0.880 − 7.95i)4-s + (3.28 − 9.02i)5-s + (3.64 + 14.2i)6-s + (−18.0 − 3.19i)7-s + (18.4 + 13.1i)8-s + (−10.2 − 24.9i)9-s + (12.8 + 23.9i)10-s + (−49.0 + 17.8i)11-s + (−36.8 − 19.1i)12-s + (−45.2 + 37.9i)13-s + (40.8 − 32.1i)14-s + (−29.4 − 40.2i)15-s + (−62.4 + 14.0i)16-s + (48.9 − 28.2i)17-s + ⋯
L(s)  = 1  + (−0.667 + 0.745i)2-s + (0.556 − 0.831i)3-s + (−0.110 − 0.993i)4-s + (0.293 − 0.807i)5-s + (0.248 + 0.968i)6-s + (−0.977 − 0.172i)7-s + (0.813 + 0.581i)8-s + (−0.381 − 0.924i)9-s + (0.405 + 0.757i)10-s + (−1.34 + 0.489i)11-s + (−0.887 − 0.461i)12-s + (−0.965 + 0.810i)13-s + (0.780 − 0.612i)14-s + (−0.507 − 0.693i)15-s + (−0.975 + 0.218i)16-s + (0.698 − 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.620 + 0.783i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.620 + 0.783i)\)
\(L(2)\)  \(\approx\)  \(0.305365 - 0.631466i\)
\(L(\frac12)\)  \(\approx\)  \(0.305365 - 0.631466i\)
\(L(\frac{5}{2})\)   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 \( 1 + (1.88 - 2.10i)T \)
3 \( 1 + (-2.88 + 4.31i)T \)
good5 \( 1 + (-3.28 + 9.02i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (18.0 + 3.19i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (49.0 - 17.8i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (45.2 - 37.9i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-48.9 + 28.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-11.8 - 6.86i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (32.1 + 182. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-199. + 237. i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (123. - 21.8i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-41.7 - 72.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-29.6 - 35.3i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (117. + 323. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (40.4 - 229. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + 110. iT - 1.48e5T^{2} \)
59 \( 1 + (-510. - 185. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-87.1 + 494. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-148. - 177. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (396. + 687. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-228. + 395. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-331. + 394. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (171. + 144. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (-425. - 245. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (799. - 290. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−13.00198578997969366071697155186, −12.14155340994714386836541816877, −10.16403829215400700297985393343, −9.460900436488932275416714577361, −8.349158999171971541974637212371, −7.36549763972387352475292231793, −6.34928172868855448364935433967, −4.91145084863215818621258863220, −2.35642339140169757935556661674, −0.42158110272763081549370388188, 2.71558934111786443225237256219, 3.32835080749240435063999786638, 5.37632132810398283335708921483, 7.32491549717029741489127110942, 8.373806552326038568744610895425, 9.754554003186657205342472910734, 10.17205349158089962965625426708, 11.03712188357831052554176341039, 12.56427927483149634978585651784, 13.43570732602625953772434270930

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