Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.812 - 0.582i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − 5.38i)2-s + (−25.9 + 18.6i)4-s − 91.5i·5-s + 158. i·7-s + (145. + 107. i)8-s + (−493 + 158. i)10-s − 580.·11-s − 166·13-s + (853. − 274. i)14-s + (327. − 970. i)16-s + 829. i·17-s + 671. i·19-s + (1.70e3 + 2.38e3i)20-s + (1.00e3 + 3.12e3i)22-s + 3.85e3·23-s + ⋯
L(s)  = 1  + (−0.306 − 0.951i)2-s + (−0.812 + 0.582i)4-s − 1.63i·5-s + 1.22i·7-s + (0.803 + 0.594i)8-s + (−1.55 + 0.501i)10-s − 1.44·11-s − 0.272·13-s + (1.16 − 0.374i)14-s + (0.320 − 0.947i)16-s + 0.695i·17-s + 0.426i·19-s + (0.954 + 1.33i)20-s + (0.442 + 1.37i)22-s + 1.52·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.812 - 0.582i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.812 - 0.582i)\)
\(L(3)\)  \(\approx\)  \(0.651789 + 0.209637i\)
\(L(\frac12)\)  \(\approx\)  \(0.651789 + 0.209637i\)
\(L(\frac{7}{2})\)   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 \( 1 + (1.73 + 5.38i)T \)
3 \( 1 \)
good5 \( 1 + 91.5iT - 3.12e3T^{2} \)
7 \( 1 - 158. iT - 1.68e4T^{2} \)
11 \( 1 + 580.T + 1.61e5T^{2} \)
13 \( 1 + 166T + 3.71e5T^{2} \)
17 \( 1 - 829. iT - 1.41e6T^{2} \)
19 \( 1 - 671. iT - 2.47e6T^{2} \)
23 \( 1 - 3.85e3T + 6.43e6T^{2} \)
29 \( 1 - 3.41e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.33e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.53e4T + 6.93e7T^{2} \)
41 \( 1 - 1.01e4iT - 1.15e8T^{2} \)
43 \( 1 + 3.00e3iT - 1.47e8T^{2} \)
47 \( 1 + 7.07e3T + 2.29e8T^{2} \)
53 \( 1 + 1.72e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.84e4T + 7.14e8T^{2} \)
61 \( 1 + 5.31e4T + 8.44e8T^{2} \)
67 \( 1 - 4.10e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.68e4T + 1.80e9T^{2} \)
73 \( 1 + 3.07e4T + 2.07e9T^{2} \)
79 \( 1 - 7.01e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.14e4T + 3.93e9T^{2} \)
89 \( 1 - 6.86e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.37e4T + 8.58e9T^{2} \)
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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

−12.75803393606929301768992890281, −12.04109533841478495276197146386, −10.78489470466492591200738167721, −9.512273713445341251802581407611, −8.702323969995413705393106513413, −7.967687086925914964170597447543, −5.46099797467672802658819417288, −4.72661546642776125940035274986, −2.80451723785192321413831739066, −1.33690246075556839003864221758, 0.30877576698172027352091276740, 2.88071851800310368833992851613, 4.57721111555965388378629222644, 6.13141339692114794077444533977, 7.35779739041483077347500254128, 7.61069747322674671738057090547, 9.532754245393054792063335209639, 10.50771011576599196460934211529, 11.07102066859651369138776038172, 13.23950826850618893533996589546

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