Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.974 - 0.222i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.28 − 2.27i)2-s + (−8.99 + 0.179i)3-s + (5.62 + 14.9i)4-s + (−2.83 − 4.90i)5-s + (29.9 + 19.9i)6-s + (45.1 + 26.0i)7-s + (15.6 − 62.0i)8-s + (80.9 − 3.23i)9-s + (−1.85 + 22.5i)10-s + (92.3 + 53.3i)11-s + (−53.3 − 133. i)12-s + (61.0 + 105. i)13-s + (−89.1 − 188. i)14-s + (26.3 + 43.6i)15-s + (−192. + 168. i)16-s − 122.·17-s + ⋯
L(s)  = 1  + (−0.822 − 0.569i)2-s + (−0.999 + 0.0199i)3-s + (0.351 + 0.936i)4-s + (−0.113 − 0.196i)5-s + (0.833 + 0.552i)6-s + (0.921 + 0.532i)7-s + (0.244 − 0.969i)8-s + (0.999 − 0.0399i)9-s + (−0.0185 + 0.225i)10-s + (0.763 + 0.440i)11-s + (−0.370 − 0.928i)12-s + (0.361 + 0.625i)13-s + (−0.454 − 0.962i)14-s + (0.117 + 0.193i)15-s + (−0.752 + 0.658i)16-s − 0.424·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.974 - 0.222i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.974 - 0.222i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.770943 + 0.0869858i\)
\(L(\frac12)\)  \(\approx\)  \(0.770943 + 0.0869858i\)
\(L(3)\)   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 \( 1 + (3.28 + 2.27i)T \)
3 \( 1 + (8.99 - 0.179i)T \)
good5 \( 1 + (2.83 + 4.90i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-45.1 - 26.0i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-92.3 - 53.3i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-61.0 - 105. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 122.T + 8.35e4T^{2} \)
19 \( 1 - 593. iT - 1.30e5T^{2} \)
23 \( 1 + (-473. + 273. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (367. - 637. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-507. + 292. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 2.28e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.43e3 - 2.48e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.94e3 + 1.12e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (913. + 527. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 4.75e3T + 7.89e6T^{2} \)
59 \( 1 + (1.86e3 - 1.07e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-33.1 + 57.4i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.55e3 + 2.05e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 5.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.70e3T + 2.83e7T^{2} \)
79 \( 1 + (1.19e3 + 690. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-2.60e3 - 1.50e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.40e3 + 4.16e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−16.28512841081192993472195924359, −14.75128621550874620006161007998, −12.72922807428983311937486378895, −11.80162614845245395945922506678, −10.99233474432385211397960407977, −9.573784102451316940128582495305, −8.166137076504341736805794023289, −6.51321883624400912634713750184, −4.43866799605159719524671703347, −1.49093064112495424875125885165, 0.952064631723440077198451708152, 4.90404189814588900605810420332, 6.46251437163803752480493181685, 7.66109417882774381316477541186, 9.294109941528985007416379995686, 11.00010884608219690074801512581, 11.26740834181866197116907549320, 13.37555889497433652502470770998, 14.87127145849355615611360008979, 15.85874615579580334350429063293

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