Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.0912 - 0.995i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.60 − 1.10i)2-s + (2.16 + 4.72i)3-s + (5.56 + 5.74i)4-s + (−0.836 + 2.29i)5-s + (−0.440 − 14.6i)6-s + (22.6 + 4.00i)7-s + (−8.16 − 21.1i)8-s + (−17.5 + 20.4i)9-s + (4.71 − 5.06i)10-s + (2.30 − 0.840i)11-s + (−15.0 + 38.7i)12-s + (−22.0 + 18.5i)13-s + (−54.7 − 35.4i)14-s + (−12.6 + 1.03i)15-s + (−1.99 + 63.9i)16-s + (−42.9 + 24.8i)17-s + ⋯
L(s)  = 1  + (−0.920 − 0.389i)2-s + (0.417 + 0.908i)3-s + (0.695 + 0.718i)4-s + (−0.0748 + 0.205i)5-s + (−0.0299 − 0.999i)6-s + (1.22 + 0.216i)7-s + (−0.360 − 0.932i)8-s + (−0.651 + 0.758i)9-s + (0.149 − 0.160i)10-s + (0.0633 − 0.0230i)11-s + (−0.362 + 0.932i)12-s + (−0.471 + 0.395i)13-s + (−1.04 − 0.676i)14-s + (−0.218 + 0.0177i)15-s + (−0.0311 + 0.999i)16-s + (−0.613 + 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0912 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0912 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.0912 - 0.995i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.0912 - 0.995i)\)
\(L(2)\)  \(\approx\)  \(0.874476 + 0.797987i\)
\(L(\frac12)\)  \(\approx\)  \(0.874476 + 0.797987i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.60 + 1.10i)T \)
3 \( 1 + (-2.16 - 4.72i)T \)
good5 \( 1 + (0.836 - 2.29i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (-22.6 - 4.00i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (-2.30 + 0.840i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (22.0 - 18.5i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (42.9 - 24.8i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-53.4 - 30.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-18.5 - 104. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (13.7 - 16.3i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (119. - 21.1i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-146. - 254. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (230. + 274. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (101. + 279. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-85.6 + 485. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 - 262. iT - 1.48e5T^{2} \)
59 \( 1 + (-811. - 295. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-64.4 + 365. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-99.5 - 118. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-224. - 388. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-393. + 681. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-624. + 744. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-268. - 225. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (-852. - 492. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.08e3 + 393. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−13.55956354353733126636339501322, −11.89390104255289820979757606427, −11.18526576316485074399658081756, −10.25713221649182608972082237232, −9.151428478700145868475574036257, −8.332665201045319147890366065262, −7.22084642893276458227170154052, −5.16919193301628309307742030156, −3.61383555880110706477935933364, −1.98077606757290208441107416189, 0.869475344100463789894139164181, 2.39370695355619180586765278885, 5.02630462838283212927101059702, 6.59714505014259922225870607251, 7.65222575485518906768495077370, 8.370882830130604257660082480084, 9.418880415124171096125668188620, 10.91842187445990625709121858840, 11.73873639169616209886536633082, 12.98913923172834001064119239955

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