Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.972 - 0.233i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.99 + 0.118i)3-s + (3.29 − 0.580i)5-s + (3.84 + 3.22i)7-s + (8.97 − 0.712i)9-s + (7.73 + 1.36i)11-s + (13.9 + 5.07i)13-s + (−9.79 + 2.13i)15-s + (18.4 − 10.6i)17-s + (−12.0 + 20.8i)19-s + (−11.9 − 9.21i)21-s + (−13.5 − 16.1i)23-s + (−12.9 + 4.72i)25-s + (−26.8 + 3.20i)27-s + (−10.7 − 29.5i)29-s + (3.12 − 2.62i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0396i)3-s + (0.658 − 0.116i)5-s + (0.549 + 0.460i)7-s + (0.996 − 0.0792i)9-s + (0.703 + 0.124i)11-s + (1.07 + 0.390i)13-s + (−0.653 + 0.142i)15-s + (1.08 − 0.625i)17-s + (−0.633 + 1.09i)19-s + (−0.567 − 0.438i)21-s + (−0.588 − 0.701i)23-s + (−0.519 + 0.189i)25-s + (−0.992 + 0.118i)27-s + (−0.370 − 1.01i)29-s + (0.100 − 0.0846i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.972 - 0.233i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (101, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.972 - 0.233i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.22542 + 0.144846i\)
\(L(\frac12)\)  \(\approx\)  \(1.22542 + 0.144846i\)
\(L(2)\)   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 \( 1 + (2.99 - 0.118i)T \)
good5 \( 1 + (-3.29 + 0.580i)T + (23.4 - 8.55i)T^{2} \)
7 \( 1 + (-3.84 - 3.22i)T + (8.50 + 48.2i)T^{2} \)
11 \( 1 + (-7.73 - 1.36i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (-13.9 - 5.07i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-18.4 + 10.6i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (12.0 - 20.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (13.5 + 16.1i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (10.7 + 29.5i)T + (-644. + 540. i)T^{2} \)
31 \( 1 + (-3.12 + 2.62i)T + (166. - 946. i)T^{2} \)
37 \( 1 + (-20.6 - 35.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (1.46 - 4.01i)T + (-1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (2.94 - 16.6i)T + (-1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-42.4 + 50.5i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 - 63.0iT - 2.80e3T^{2} \)
59 \( 1 + (75.1 - 13.2i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (76.5 + 64.2i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-83.0 - 30.2i)T + (3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (-4.34 + 2.50i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (19.2 - 33.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (35.9 - 13.0i)T + (4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (48.8 + 134. i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-75.2 - 43.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-28.6 + 162. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−13.46236939825087796647917970976, −12.19243008548941353242340550943, −11.57797763448419832868681434585, −10.36657725328666221806707639856, −9.405298368675168366546659243176, −7.997199109415274856179178652537, −6.36372384954382458500755790556, −5.64174078577450024858746030262, −4.17213890365882923782834480024, −1.58170561685035816215127035065, 1.37142095806791880079061842747, 3.99125076216812481810468216655, 5.52502632154236091553407405334, 6.41676746985790358593438896770, 7.76238786270979964209947859910, 9.292905976659324101230159227115, 10.53398945837924193091564459993, 11.15301907776253136009865522063, 12.33719059789091914724670926764, 13.37795531059924456270538168871

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