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Properties

Label 2-363-3.2-c2-0-25
Degree $2$
Conductor $363$
Sign $-0.0620 - 0.998i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.792i·2-s + (0.186 + 2.99i)3-s + 3.37·4-s − 2.52i·5-s + (−2.37 + 0.147i)6-s + 4.74·7-s + 5.84i·8-s + (−8.93 + 1.11i)9-s + 2·10-s + (0.627 + 10.0i)12-s + 13.4·13-s + 3.75i·14-s + (7.55 − 0.469i)15-s + 8.86·16-s + 22.6i·17-s + (−0.883 − 7.07i)18-s + ⋯
L(s)  = 1  + 0.396i·2-s + (0.0620 + 0.998i)3-s + 0.843·4-s − 0.504i·5-s + (−0.395 + 0.0245i)6-s + 0.677·7-s + 0.730i·8-s + (−0.992 + 0.123i)9-s + 0.200·10-s + (0.0523 + 0.841i)12-s + 1.03·13-s + 0.268i·14-s + (0.503 − 0.0313i)15-s + 0.553·16-s + 1.33i·17-s + (−0.0490 − 0.393i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0620 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0620 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.0620 - 0.998i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.0620 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.53629 + 1.63476i\)
\(L(\frac12)\) \(\approx\) \(1.53629 + 1.63476i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.186 - 2.99i)T \)
11 \( 1 \)
good2 \( 1 - 0.792iT - 4T^{2} \)
5 \( 1 + 2.52iT - 25T^{2} \)
7 \( 1 - 4.74T + 49T^{2} \)
13 \( 1 - 13.4T + 169T^{2} \)
17 \( 1 - 22.6iT - 289T^{2} \)
19 \( 1 + 8.23T + 361T^{2} \)
23 \( 1 - 30.2iT - 529T^{2} \)
29 \( 1 + 53.6iT - 841T^{2} \)
31 \( 1 - 25.8T + 961T^{2} \)
37 \( 1 + 42.6T + 1.36e3T^{2} \)
41 \( 1 - 30.8iT - 1.68e3T^{2} \)
43 \( 1 + 35.4T + 1.84e3T^{2} \)
47 \( 1 - 31.2iT - 2.20e3T^{2} \)
53 \( 1 + 9.91iT - 2.80e3T^{2} \)
59 \( 1 + 61.5iT - 3.48e3T^{2} \)
61 \( 1 + 51.4T + 3.72e3T^{2} \)
67 \( 1 - 34.3T + 4.48e3T^{2} \)
71 \( 1 + 14.5iT - 5.04e3T^{2} \)
73 \( 1 - 88.2T + 5.32e3T^{2} \)
79 \( 1 - 93.6T + 6.24e3T^{2} \)
83 \( 1 + 34.1iT - 6.88e3T^{2} \)
89 \( 1 + 143. iT - 7.92e3T^{2} \)
97 \( 1 + 40.2T + 9.40e3T^{2} \)
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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

−11.24439087719795318185879162014, −10.69495830114974227308268170749, −9.641149661823372532177834738458, −8.351561668824258403417238958317, −8.105115383201316676688107673049, −6.44151999897873301883755777738, −5.64658660148817319180914448752, −4.56295122647449617063353668300, −3.37727888797820370120226749668, −1.72738474262433880498118009973, 1.10750588875625658280892590154, 2.34501074048050583273722498603, 3.33654764919392249792525378454, 5.17980139884944245972525199799, 6.58937285971004447660358179237, 6.91415010881811839049358458556, 8.080784392590254856110178216534, 8.938222952913438666405644172887, 10.57565510305441790085341651228, 10.96764306886227379141224596546

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