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Properties

Label 2-363-11.9-c1-0-14
Degree $2$
Conductor $363$
Sign $-0.577 + 0.816i$
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  + (0.535 − 1.64i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.927 − 2.85i)5-s + (−0.535 − 1.64i)6-s + (2.80 + 2.03i)7-s + (1.40 − 1.01i)8-s + (0.309 − 0.951i)9-s − 5.19·10-s − 0.999·12-s + (−0.535 + 1.64i)13-s + (4.85 − 3.52i)14-s + (−2.42 − 1.76i)15-s + (−1.54 − 4.75i)16-s + (0.535 + 1.64i)17-s + (−1.40 − 1.01i)18-s + ⋯
L(s)  = 1  + (0.378 − 1.16i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.414 − 1.27i)5-s + (−0.218 − 0.672i)6-s + (1.05 + 0.769i)7-s + (0.495 − 0.359i)8-s + (0.103 − 0.317i)9-s − 1.64·10-s − 0.288·12-s + (−0.148 + 0.456i)13-s + (1.29 − 0.942i)14-s + (−0.626 − 0.455i)15-s + (−0.386 − 1.18i)16-s + (0.129 + 0.399i)17-s + (−0.330 − 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.926172 - 1.78874i\)
\(L(\frac12)\) \(\approx\) \(0.926172 - 1.78874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.535 + 1.64i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.927 + 2.85i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.80 - 2.03i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.535 - 1.64i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.535 - 1.64i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.60 - 4.07i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-1.40 - 1.01i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.23 + 3.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.89 - 6.46i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.78 - 8.55i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.85 - 3.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (1.85 + 5.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.60 + 4.07i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (2.16 - 6.65i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.49649722059092517567676050119, −10.36472215565711527994248399896, −9.255216130614050327366296639752, −8.319502316688747609015933021903, −7.80088092896954727409020780334, −6.04473797560603592611835847145, −4.64389207856629246177084570818, −4.05061497994884935023729049188, −2.35394230324454250808928624287, −1.43492699702706317149551454200, 2.40444172513942350548236196685, 3.97713371202421931921103275932, 4.80527792630616643529422370757, 6.15827007181814166542170597300, 7.14065158232685482207453373692, 7.72405983416698438555192785608, 8.493292356411723490582523298062, 10.08502891966320945913348904617, 10.87346241444923413655565456279, 11.37701627675583626168566167010

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