Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.859 + 0.511i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.93 − 0.717i)2-s + (2.22 − 8.72i)3-s + (14.9 + 5.64i)4-s + (−5.51 − 9.55i)5-s + (−14.9 + 32.7i)6-s + (−10.3 − 5.95i)7-s + (−54.8 − 32.9i)8-s + (−71.1 − 38.7i)9-s + (14.8 + 41.5i)10-s + (−189. − 109. i)11-s + (82.4 − 118. i)12-s + (18.5 + 32.1i)13-s + (36.3 + 30.8i)14-s + (−95.5 + 26.8i)15-s + (192. + 169. i)16-s + 284.·17-s + ⋯
L(s)  = 1  + (−0.983 − 0.179i)2-s + (0.246 − 0.969i)3-s + (0.935 + 0.352i)4-s + (−0.220 − 0.382i)5-s + (−0.416 + 0.909i)6-s + (−0.210 − 0.121i)7-s + (−0.857 − 0.514i)8-s + (−0.878 − 0.478i)9-s + (0.148 + 0.415i)10-s + (−1.57 − 0.906i)11-s + (0.572 − 0.819i)12-s + (0.109 + 0.189i)13-s + (0.185 + 0.157i)14-s + (−0.424 + 0.119i)15-s + (0.751 + 0.660i)16-s + 0.982·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.859 + 0.511i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.859 + 0.511i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.181870 - 0.660944i\)
\(L(\frac12)\)  \(\approx\)  \(0.181870 - 0.660944i\)
\(L(3)\)   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 \( 1 + (3.93 + 0.717i)T \)
3 \( 1 + (-2.22 + 8.72i)T \)
good5 \( 1 + (5.51 + 9.55i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (10.3 + 5.95i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (189. + 109. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-18.5 - 32.1i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 284.T + 8.35e4T^{2} \)
19 \( 1 + 45.4iT - 1.30e5T^{2} \)
23 \( 1 + (174. - 100. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-614. + 1.06e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-1.31e3 + 757. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.52e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.31e3 - 2.28e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-34.6 - 19.9i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.49e3 - 1.44e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.41e3T + 7.89e6T^{2} \)
59 \( 1 + (2.45e3 - 1.41e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.62e3 + 4.55e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (805. - 465. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.16e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.16e3T + 2.83e7T^{2} \)
79 \( 1 + (6.48e3 + 3.74e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-966. - 558. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 6.73e3T + 6.27e7T^{2} \)
97 \( 1 + (6.02e3 - 1.04e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.56410143723863557203575460370, −13.78581099157121697254601391574, −12.62883955128335421707474963498, −11.53655704748609697205643634594, −10.09776298803310237228067381084, −8.439813587776855557796802842807, −7.74076724297726126359626348488, −6.08389257144491309383063868405, −2.78739624084413339462805373706, −0.61499935531518778601015653380, 2.88040522483510262610113451517, 5.35408802504150532748027408245, 7.37686596770825933757417174074, 8.632474919959903036220789550413, 10.11356772001492129881430158992, 10.62294983757387605062269948465, 12.27719224321429801302189732653, 14.30977228828919028763342892515, 15.47092444118170318057471116131, 15.97311362501348451950870450407

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