Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.227 - 0.973i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.343 + 1.97i)2-s + (2.96 − 0.463i)3-s + (−3.76 + 1.35i)4-s + (0.608 + 3.44i)5-s + (1.93 + 5.68i)6-s + (−5.71 + 6.81i)7-s + (−3.96 − 6.94i)8-s + (8.56 − 2.74i)9-s + (−6.58 + 2.38i)10-s + (5.38 + 0.949i)11-s + (−10.5 + 5.76i)12-s + (21.3 + 7.76i)13-s + (−15.3 − 8.92i)14-s + (3.40 + 9.93i)15-s + (12.3 − 10.1i)16-s + (−8.12 − 14.0i)17-s + ⋯
L(s)  = 1  + (0.171 + 0.985i)2-s + (0.987 − 0.154i)3-s + (−0.940 + 0.338i)4-s + (0.121 + 0.689i)5-s + (0.322 + 0.946i)6-s + (−0.816 + 0.973i)7-s + (−0.495 − 0.868i)8-s + (0.952 − 0.305i)9-s + (−0.658 + 0.238i)10-s + (0.489 + 0.0862i)11-s + (−0.877 + 0.480i)12-s + (1.64 + 0.597i)13-s + (−1.09 − 0.637i)14-s + (0.226 + 0.662i)15-s + (0.770 − 0.637i)16-s + (−0.478 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.227 - 0.973i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.227 - 0.973i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.09517 + 1.38028i\)
\(L(\frac12)\)  \(\approx\)  \(1.09517 + 1.38028i\)
\(L(2)\)   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 \( 1 + (-0.343 - 1.97i)T \)
3 \( 1 + (-2.96 + 0.463i)T \)
good5 \( 1 + (-0.608 - 3.44i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (5.71 - 6.81i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (-5.38 - 0.949i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (-21.3 - 7.76i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (8.12 + 14.0i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (19.5 + 11.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (22.1 + 26.4i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-24.1 + 8.78i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-14.3 - 17.1i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-7.88 - 13.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (49.6 + 18.0i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-9.61 - 1.69i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-14.9 + 17.8i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 - 4.09T + 2.80e3T^{2} \)
59 \( 1 + (-28.0 + 4.93i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (31.1 + 26.1i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (30.9 - 84.9i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (87.8 - 50.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (16.0 - 27.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (8.65 + 23.7i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (16.3 + 45.0i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-45.0 + 78.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (25.7 - 146. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−13.91136249720335323190074503316, −13.15663538202639855616534219482, −12.02809468530421954491994504491, −10.22118329152754693141249600593, −8.927461144492627027842092993428, −8.549728280853311019118596917557, −6.74348302508624172339038059338, −6.35797629884059476607848095326, −4.22474325001081770623190994117, −2.81200729798151428766010025971, 1.40213673721402418134861756102, 3.47346504889288503578875408402, 4.21671165451971749586719865086, 6.19300944669388961469781630473, 8.166357527416623923337686818739, 8.944908799380637311383095308916, 10.05754341715908385126264005560, 10.80212407696473637092996247497, 12.42098087714253506872856344944, 13.36187213377320890142732928475

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