Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.643 - 0.765i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.86 + 1.03i)2-s + (7.79 + 4.49i)3-s + (13.8 − 7.96i)4-s + (−5.89 − 10.2i)5-s + (−34.7 − 9.32i)6-s + (50.5 + 29.1i)7-s + (−45.4 + 45.0i)8-s + (40.6 + 70.0i)9-s + (33.2 + 33.3i)10-s + (86.9 + 50.2i)11-s + (143. + 0.228i)12-s + (85.3 + 147. i)13-s + (−225. − 60.7i)14-s + (−0.116 − 106. i)15-s + (129. − 221. i)16-s − 398.·17-s + ⋯
L(s)  = 1  + (−0.966 + 0.257i)2-s + (0.866 + 0.499i)3-s + (0.867 − 0.497i)4-s + (−0.235 − 0.408i)5-s + (−0.965 − 0.259i)6-s + (1.03 + 0.595i)7-s + (−0.709 + 0.704i)8-s + (0.501 + 0.864i)9-s + (0.332 + 0.333i)10-s + (0.718 + 0.414i)11-s + (0.999 + 0.00158i)12-s + (0.504 + 0.874i)13-s + (−1.15 − 0.309i)14-s + (−0.000517 − 0.471i)15-s + (0.504 − 0.863i)16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.643 - 0.765i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.643 - 0.765i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.19272 + 0.555821i\)
\(L(\frac12)\)  \(\approx\)  \(1.19272 + 0.555821i\)
\(L(3)\)   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 \( 1 + (3.86 - 1.03i)T \)
3 \( 1 + (-7.79 - 4.49i)T \)
good5 \( 1 + (5.89 + 10.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-50.5 - 29.1i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-86.9 - 50.2i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-85.3 - 147. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 398.T + 8.35e4T^{2} \)
19 \( 1 + 404. iT - 1.30e5T^{2} \)
23 \( 1 + (-291. + 168. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-327. + 567. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (550. - 317. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.59e3T + 1.87e6T^{2} \)
41 \( 1 + (1.23e3 + 2.13e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.93e3 + 1.11e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.51e3 - 1.45e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.29e3T + 7.89e6T^{2} \)
59 \( 1 + (1.00e3 - 578. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.96e3 - 5.12e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.08e3 + 1.78e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 5.63e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.49e3T + 2.83e7T^{2} \)
79 \( 1 + (2.78e3 + 1.61e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-7.06e3 - 4.07e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 910.T + 6.27e7T^{2} \)
97 \( 1 + (-8.80e3 + 1.52e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.73678465536184769258221955973, −15.06009265050372830187801114880, −13.87549220737690383553135388310, −11.86436164622771938399505783082, −10.72243503968282509690403772055, −8.907346082052716262318752403804, −8.740018039758421767673087348522, −6.99699071535816337728236894696, −4.67417646424293662494080466881, −2.01099162173915924132802087843, 1.44019910810240842522800285201, 3.48446239304172014491262638343, 6.78237528680141657663360196990, 7.951416305557631619558025669473, 8.891109320190526872724151311038, 10.55068969472636774058930832833, 11.58524673458842026088429790136, 13.12050499154512022333426531122, 14.48398410853910418434070996804, 15.46258467325682160262874172293

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