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Properties

Label 2-363-11.3-c1-0-0
Degree $2$
Conductor $363$
Sign $-0.220 - 0.975i$
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  + (−1.61 + 1.17i)2-s + (−0.309 − 0.951i)3-s + (0.618 − 1.90i)4-s + (−3.23 − 2.35i)5-s + (1.61 + 1.17i)6-s + (−0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + 8·10-s − 2.00·12-s + (−1.61 + 1.17i)13-s + (−0.618 − 1.90i)14-s + (−1.23 + 3.80i)15-s + (3.23 + 2.35i)16-s + (3.23 + 2.35i)17-s + (0.618 − 1.90i)18-s + (0.927 + 2.85i)19-s + ⋯
L(s)  = 1  + (−1.14 + 0.831i)2-s + (−0.178 − 0.549i)3-s + (0.309 − 0.951i)4-s + (−1.44 − 1.05i)5-s + (0.660 + 0.479i)6-s + (−0.116 + 0.359i)7-s + (−0.269 + 0.195i)9-s + 2.52·10-s − 0.577·12-s + (−0.448 + 0.326i)13-s + (−0.165 − 0.508i)14-s + (−0.319 + 0.982i)15-s + (0.809 + 0.587i)16-s + (0.784 + 0.570i)17-s + (0.145 − 0.448i)18-s + (0.212 + 0.654i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196893 + 0.246451i\)
\(L(\frac12)\) \(\approx\) \(0.196893 + 0.246451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (1.61 - 1.17i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (3.23 + 2.35i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.61 - 1.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.23 - 2.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.927 - 2.85i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.04 + 2.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.618 - 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-0.618 - 1.90i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.85 - 3.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.09 - 9.51i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.42 - 1.76i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.39 + 10.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.89 + 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.85 - 3.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (4.04 - 2.93i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.93032401979315222135830036141, −10.65058642290770805420711172272, −9.410876080199593316194690741864, −8.672465338921912355518080534509, −7.88737758257977206495171249189, −7.42244813729472633909546977854, −6.23154485786090029282553865122, −5.02197415486427816098734713618, −3.61281034113458789604522528445, −1.15785176556516755631409637777, 0.38443575449745238907659246676, 2.79451118025348070337925609331, 3.59716559281281209627703992637, 5.00891944400626884196175299346, 6.77916747609704950104615633233, 7.68927421985911301835714850372, 8.414590856015217969918428000317, 9.702893939266454337267362554147, 10.23602852073065177262488462202, 11.14001622986651992671253159714

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