Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.481 - 0.876i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.89 + 0.903i)2-s + (−8.98 + 0.573i)3-s + (14.3 + 7.03i)4-s + (19.5 + 33.8i)5-s + (−35.5 − 5.87i)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 + (−133. − 54.9i)12-s + (−104. − 180. i)13-s + (35.6 + 33.3i)14-s + (−194. − 292. i)15-s + (156. + 202. i)16-s + 93.3·17-s + ⋯
L(s)  = 1  + (0.974 + 0.225i)2-s + (−0.997 + 0.0637i)3-s + (0.898 + 0.439i)4-s + (0.781 + 1.35i)5-s + (−0.986 − 0.163i)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 + (−0.924 − 0.381i)12-s + (−0.618 − 1.07i)13-s + (0.182 + 0.170i)14-s + (−0.866 − 1.30i)15-s + (0.613 + 0.790i)16-s + 0.323·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.481 - 0.876i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.481 - 0.876i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.78628 + 1.05675i\)
\(L(\frac12)\)  \(\approx\)  \(1.78628 + 1.05675i\)
\(L(3)\)   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 \( 1 + (-3.89 - 0.903i)T \)
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

−15.64084250807008376281850183291, −14.76709950431556681148070725319, −13.52920486900602567125233891951, −12.38465798596275046433172123551, −10.93385177980487679007102546229, −10.35820195810813945440851557153, −7.45736719725082984148440693352, −6.23395398175269309527035179044, −5.16859912675321878792583339067, −2.81569492335073302967713943166, 1.53990652692313975715810553501, 4.72872415354178031263414269603, 5.43154964849652367823132230071, 7.10972336288007035299024237659, 9.493203359543124633508545769770, 10.82541225492853380505700101749, 12.22770572513788439698503599653, 12.84111828992241749560511664982, 13.98565855964266879751865852555, 15.63195988603139402278916206381

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