Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.993 + 0.114i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.73 − 2.92i)2-s + (8.98 − 0.573i)3-s + (−1.08 + 15.9i)4-s + (19.5 + 33.8i)5-s + (−26.2 − 24.6i)6-s + (−10.5 − 6.10i)7-s + (49.6 − 40.4i)8-s + (80.3 − 10.3i)9-s + (45.5 − 149. i)10-s + (96.1 + 55.5i)11-s + (−0.620 + 143. i)12-s + (−104. − 180. i)13-s + (11.0 + 47.5i)14-s + (194. + 292. i)15-s + (−253. − 34.7i)16-s + 93.3·17-s + ⋯
L(s)  = 1  + (−0.682 − 0.730i)2-s + (0.997 − 0.0637i)3-s + (−0.0680 + 0.997i)4-s + (0.781 + 1.35i)5-s + (−0.727 − 0.685i)6-s + (−0.215 − 0.124i)7-s + (0.775 − 0.631i)8-s + (0.991 − 0.127i)9-s + (0.455 − 1.49i)10-s + (0.794 + 0.458i)11-s + (−0.00430 + 0.999i)12-s + (−0.618 − 1.07i)13-s + (0.0562 + 0.242i)14-s + (0.866 + 1.30i)15-s + (−0.990 − 0.135i)16-s + 0.323·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.993 + 0.114i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.993 + 0.114i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.52645 - 0.0878845i\)
\(L(\frac12)\)  \(\approx\)  \(1.52645 - 0.0878845i\)
\(L(3)\)   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 \( 1 + (2.73 + 2.92i)T \)
3 \( 1 + (-8.98 + 0.573i)T \)
good5 \( 1 + (-19.5 - 33.8i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (10.5 + 6.10i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-96.1 - 55.5i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (104. + 180. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 93.3T + 8.35e4T^{2} \)
19 \( 1 - 26.8iT - 1.30e5T^{2} \)
23 \( 1 + (757. - 437. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-650. + 1.12e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-593. + 342. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.76e3T + 1.87e6T^{2} \)
41 \( 1 + (39.0 + 67.6i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.40e3 + 811. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.99e3 + 1.15e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.31e3T + 7.89e6T^{2} \)
59 \( 1 + (-4.81e3 + 2.78e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.09e3 - 1.88e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (213. - 123. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.60e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.56e3T + 2.83e7T^{2} \)
79 \( 1 + (-4.48e3 - 2.59e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (1.62e3 + 936. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.16e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.86e3 + 4.97e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.51710188661039418099674570556, −14.32585563036561009905929967680, −13.37639738514091342452768133179, −11.92857068752945045647293397578, −10.09378354711797320318325735365, −9.868264769440402610405648391675, −8.043031198929880560017810175427, −6.80040139645891562072545533328, −3.49761277219990896921751307853, −2.16089593979534874751017280059, 1.58555972493759118783091986684, 4.74027032902030097027732069730, 6.54724302431915538665266809438, 8.353776524932909533908075284668, 9.126412544974546791382450814332, 9.995722384168523335137885513430, 12.32231540858047032272079675639, 13.81109616693419695647326588708, 14.42151674695449766673322863160, 16.08017712253999851093859966512

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