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Properties

Label 2-363-11.9-c1-0-7
Degree $2$
Conductor $363$
Sign $0.719 - 0.694i$
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  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (−0.618 − 1.90i)5-s + (0.309 + 0.951i)6-s + (3.23 + 2.35i)7-s + (−2.42 + 1.76i)8-s + (0.309 − 0.951i)9-s + 1.99·10-s + 12-s + (0.618 − 1.90i)13-s + (−3.23 + 2.35i)14-s + (−1.61 − 1.17i)15-s + (−0.309 − 0.951i)16-s + (0.618 + 1.90i)17-s + (0.809 + 0.587i)18-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (0.404 + 0.293i)4-s + (−0.276 − 0.850i)5-s + (0.126 + 0.388i)6-s + (1.22 + 0.888i)7-s + (−0.858 + 0.623i)8-s + (0.103 − 0.317i)9-s + 0.632·10-s + 0.288·12-s + (0.171 − 0.527i)13-s + (−0.864 + 0.628i)14-s + (−0.417 − 0.303i)15-s + (−0.0772 − 0.237i)16-s + (0.149 + 0.461i)17-s + (0.190 + 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.719 - 0.694i$
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: \((2,\ 363,\ (\ :1/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53473 + 0.619573i\)
\(L(\frac12)\) \(\approx\) \(1.53473 + 0.619573i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.618 + 1.90i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.23 - 2.35i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.618 + 1.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.618 - 1.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (4.85 + 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.47 - 7.60i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.85 + 3.52i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.61 - 1.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (6.47 - 4.70i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.85 + 5.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.23 - 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.85 + 5.70i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (11.3 + 8.22i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.23 + 3.80i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.70 + 11.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-0.618 + 1.90i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.67805846077870929938728622217, −10.81649705952253333475672343055, −9.077796458767855077568000040159, −8.567982290302422934725276694513, −7.959097778485646671065223030361, −7.04087804897277493968612192947, −5.72847286849391974811638868002, −4.87407931774073950344897226345, −3.15610091110618463559268987797, −1.71684100947297259348320137401, 1.49633516232260861168588190176, 2.87592822451911397101739816204, 3.94845719117174148511132884871, 5.25280233912964627661951953305, 6.89623714302129936880438135400, 7.40874187366503980935469863211, 8.708445421855586774809953100062, 9.695362117651877324787003615298, 10.68630404786237188350348878147, 11.10211445040991423154753015562

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