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  :  $\chi_{36} (31, \cdot )$
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

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

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