Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.999 + 0.0374i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 1.68i)2-s + (2.99 − 0.107i)3-s + (−1.69 − 3.62i)4-s + (−0.755 − 4.28i)5-s + (−3.03 + 5.17i)6-s + (6.44 − 7.67i)7-s + (7.93 + 1.04i)8-s + (8.97 − 0.645i)9-s + (8.03 + 3.32i)10-s + (−16.5 − 2.92i)11-s + (−5.46 − 10.6i)12-s + (7.70 + 2.80i)13-s + (6.02 + 19.1i)14-s + (−2.72 − 12.7i)15-s + (−10.2 + 12.2i)16-s + (13.2 + 22.9i)17-s + ⋯
L(s)  = 1  + (−0.537 + 0.843i)2-s + (0.999 − 0.0358i)3-s + (−0.422 − 0.906i)4-s + (−0.151 − 0.856i)5-s + (−0.506 + 0.862i)6-s + (0.920 − 1.09i)7-s + (0.991 + 0.130i)8-s + (0.997 − 0.0717i)9-s + (0.803 + 0.332i)10-s + (−1.50 − 0.265i)11-s + (−0.455 − 0.890i)12-s + (0.592 + 0.215i)13-s + (0.430 + 1.36i)14-s + (−0.181 − 0.850i)15-s + (−0.642 + 0.766i)16-s + (0.780 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0374i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.999 + 0.0374i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.999 + 0.0374i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.40379 - 0.0263046i\)
\(L(\frac12)\)  \(\approx\)  \(1.40379 - 0.0263046i\)
\(L(2)\)   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 \( 1 + (1.07 - 1.68i)T \)
3 \( 1 + (-2.99 + 0.107i)T \)
good5 \( 1 + (0.755 + 4.28i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-6.44 + 7.67i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (16.5 + 2.92i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (-7.70 - 2.80i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-13.2 - 22.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (10.2 + 5.92i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-14.3 - 17.1i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (29.1 - 10.6i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (12.9 + 15.4i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-20.1 - 34.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (34.1 + 12.4i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (25.3 + 4.46i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-12.9 + 15.4i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 - 46.7T + 2.80e3T^{2} \)
59 \( 1 + (36.4 - 6.42i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-48.7 - 40.9i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (25.9 - 71.3i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-17.4 + 10.0i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (50.2 - 87.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-29.7 - 81.8i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (25.2 + 69.3i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (37.6 - 65.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-12.6 + 71.5i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−13.46761766035821176498092636983, −13.06090599466685445671292015083, −10.91823501788423225277364717278, −10.08723138529608615026875427923, −8.641195828665997133082495321068, −8.124168664372969269450255618973, −7.23764546025445204074670202693, −5.33252854787902580789152742140, −4.09800192771930087747627055487, −1.36432018207655545816495676382, 2.22786000635407601604661678810, 3.15790058006176745071726076032, 5.00692380236851033173178470541, 7.38586440377361617816325243036, 8.161831889376364420299544109942, 9.139129641607869154577275682829, 10.34860761338922249493313296468, 11.15626125910359477233429066169, 12.38742165837509185606297933553, 13.34008798680484106704951179670

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