Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.942 - 0.334i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.88 + 0.812i)3-s + (0.980 + 1.16i)5-s + (3.23 − 1.17i)7-s + (7.67 + 4.69i)9-s + (−3.21 + 3.82i)11-s + (−0.778 − 4.41i)13-s + (1.88 + 4.17i)15-s + (−3.57 + 2.06i)17-s + (6.75 − 11.7i)19-s + (10.3 − 0.772i)21-s + (−5.79 + 15.9i)23-s + (3.93 − 22.3i)25-s + (18.3 + 19.7i)27-s + (−47.1 − 8.30i)29-s + (−14.3 − 5.22i)31-s + ⋯
L(s)  = 1  + (0.962 + 0.270i)3-s + (0.196 + 0.233i)5-s + (0.462 − 0.168i)7-s + (0.853 + 0.521i)9-s + (−0.291 + 0.347i)11-s + (−0.0599 − 0.339i)13-s + (0.125 + 0.278i)15-s + (−0.210 + 0.121i)17-s + (0.355 − 0.615i)19-s + (0.491 − 0.0367i)21-s + (−0.252 + 0.692i)23-s + (0.157 − 0.893i)25-s + (0.680 + 0.733i)27-s + (−1.62 − 0.286i)29-s + (−0.462 − 0.168i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.942 - 0.334i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.942 - 0.334i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.84449 + 0.317250i\)
\(L(\frac12)\)  \(\approx\)  \(1.84449 + 0.317250i\)
\(L(2)\)   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 \( 1 + (-2.88 - 0.812i)T \)
good5 \( 1 + (-0.980 - 1.16i)T + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (-3.23 + 1.17i)T + (37.5 - 31.4i)T^{2} \)
11 \( 1 + (3.21 - 3.82i)T + (-21.0 - 119. i)T^{2} \)
13 \( 1 + (0.778 + 4.41i)T + (-158. + 57.8i)T^{2} \)
17 \( 1 + (3.57 - 2.06i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.75 + 11.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (5.79 - 15.9i)T + (-405. - 340. i)T^{2} \)
29 \( 1 + (47.1 + 8.30i)T + (790. + 287. i)T^{2} \)
31 \( 1 + (14.3 + 5.22i)T + (736. + 617. i)T^{2} \)
37 \( 1 + (32.3 + 56.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (55.4 - 9.76i)T + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (-22.7 - 19.0i)T + (321. + 1.82e3i)T^{2} \)
47 \( 1 + (7.04 + 19.3i)T + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 + 19.8iT - 2.80e3T^{2} \)
59 \( 1 + (-63.6 - 75.9i)T + (-604. + 3.42e3i)T^{2} \)
61 \( 1 + (-77.5 + 28.2i)T + (2.85e3 - 2.39e3i)T^{2} \)
67 \( 1 + (-11.2 - 63.7i)T + (-4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (-109. + 63.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-18.0 + 31.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (14.4 - 82.1i)T + (-5.86e3 - 2.13e3i)T^{2} \)
83 \( 1 + (16.5 + 2.91i)T + (6.47e3 + 2.35e3i)T^{2} \)
89 \( 1 + (-66.0 - 38.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-82.1 - 68.9i)T + (1.63e3 + 9.26e3i)T^{2} \)
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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

−13.63361960857972212927145520074, −12.75088610665459052692092916520, −11.25808143416264390947548778710, −10.20663007001672647677499985586, −9.236474177103108031915304470255, −8.054359842723758777633652211078, −7.09909253637948803491141467294, −5.24755113741049135973818483433, −3.78685313200206169813840859474, −2.17777231796777701186422145520, 1.84102769588108979564286071308, 3.55219831524948682524962395281, 5.21326135508421548416794106201, 6.87042163097813747884371921444, 8.066699056910992685855513005810, 8.940453468885260607868061260806, 10.01265812654839720307539157072, 11.38549854991031435064195609625, 12.58507263798865248296985372965, 13.48196893112784938829081487272

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