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Properties

Label 2-363-33.29-c1-0-15
Degree $2$
Conductor $363$
Sign $0.999 - 0.00245i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + 0.0369i)3-s + (−0.618 − 1.90i)4-s + (1.94 + 2.68i)5-s + (2.99 + 0.127i)9-s + (−1.00 − 3.31i)12-s + (3.27 + 4.71i)15-s + (−3.23 + 2.35i)16-s + (3.89 − 5.36i)20-s − 3.31i·23-s + (−1.85 + 5.70i)25-s + (5.18 + 0.332i)27-s + (−4.04 − 2.93i)31-s + (−1.60 − 5.78i)36-s + (−2.16 − 6.65i)37-s + (5.5 + 8.29i)45-s + ⋯
L(s)  = 1  + (0.999 + 0.0213i)3-s + (−0.309 − 0.951i)4-s + (0.871 + 1.19i)5-s + (0.999 + 0.0426i)9-s + (−0.288 − 0.957i)12-s + (0.846 + 1.21i)15-s + (−0.809 + 0.587i)16-s + (0.871 − 1.19i)20-s − 0.691i·23-s + (−0.370 + 1.14i)25-s + (0.997 + 0.0639i)27-s + (−0.726 − 0.527i)31-s + (−0.268 − 0.963i)36-s + (−0.355 − 1.09i)37-s + (0.819 + 1.23i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.999 - 0.00245i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.999 - 0.00245i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96727 + 0.00241721i\)
\(L(\frac12)\) \(\approx\) \(1.96727 + 0.00241721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0369i)T \)
11 \( 1 \)
good2 \( 1 + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-1.94 - 2.68i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.04 + 2.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.16 + 6.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-6.30 - 2.04i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.79 - 10.7i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.15 + 1.02i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (9.74 + 13.4i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (13.7 + 9.99i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92557522852007777108435124571, −10.45532342335344406901487198217, −9.572923234385192446617241250286, −8.972235739481097782651172717127, −7.59850067552333734834300071716, −6.61735430692419906962892629144, −5.74810479397788876624289177103, −4.33423142860740175041828266427, −2.89585754527829819190238181197, −1.82600190367722096371917388635, 1.70525901996948540872432637548, 3.14335217985389202799248693957, 4.32426539416367784376703319195, 5.32542213950031770878289821634, 6.90191672448881386440399631100, 7.976943649312448519788434191628, 8.715658090476225456710677984403, 9.301700886132544643448180072750, 10.12918803456573230977133377116, 11.71034042738540202196516254755

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