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Properties

Label 2-363-3.2-c2-0-50
Degree $2$
Conductor $363$
Sign $-0.994 + 0.103i$
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  − 1.89i·2-s + (−2.98 + 0.311i)3-s + 0.398·4-s − 8.25i·5-s + (0.592 + 5.66i)6-s + 3.60·7-s − 8.34i·8-s + (8.80 − 1.86i)9-s − 15.6·10-s + (−1.18 + 0.124i)12-s + 13.1·13-s − 6.84i·14-s + (2.57 + 24.6i)15-s − 14.2·16-s − 25.6i·17-s + (−3.53 − 16.7i)18-s + ⋯
L(s)  = 1  − 0.948i·2-s + (−0.994 + 0.103i)3-s + 0.0995·4-s − 1.65i·5-s + (0.0986 + 0.943i)6-s + 0.515·7-s − 1.04i·8-s + (0.978 − 0.206i)9-s − 1.56·10-s + (−0.0989 + 0.0103i)12-s + 1.00·13-s − 0.488i·14-s + (0.171 + 1.64i)15-s − 0.890·16-s − 1.50i·17-s + (−0.196 − 0.928i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.994 + 0.103i$
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: \((2,\ 363,\ (\ :1),\ -0.994 + 0.103i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0731102 - 1.40227i\)
\(L(\frac12)\) \(\approx\) \(0.0731102 - 1.40227i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.98 - 0.311i)T \)
11 \( 1 \)
good2 \( 1 + 1.89iT - 4T^{2} \)
5 \( 1 + 8.25iT - 25T^{2} \)
7 \( 1 - 3.60T + 49T^{2} \)
13 \( 1 - 13.1T + 169T^{2} \)
17 \( 1 + 25.6iT - 289T^{2} \)
19 \( 1 - 12.5T + 361T^{2} \)
23 \( 1 - 27.7iT - 529T^{2} \)
29 \( 1 + 9.48iT - 841T^{2} \)
31 \( 1 + 5.08T + 961T^{2} \)
37 \( 1 + 34.6T + 1.36e3T^{2} \)
41 \( 1 - 34.2iT - 1.68e3T^{2} \)
43 \( 1 + 39.5T + 1.84e3T^{2} \)
47 \( 1 - 19.4iT - 2.20e3T^{2} \)
53 \( 1 + 17.2iT - 2.80e3T^{2} \)
59 \( 1 + 27.1iT - 3.48e3T^{2} \)
61 \( 1 - 43.7T + 3.72e3T^{2} \)
67 \( 1 - 82.0T + 4.48e3T^{2} \)
71 \( 1 - 117. iT - 5.04e3T^{2} \)
73 \( 1 - 107.T + 5.32e3T^{2} \)
79 \( 1 - 62.8T + 6.24e3T^{2} \)
83 \( 1 + 49.1iT - 6.88e3T^{2} \)
89 \( 1 - 69.1iT - 7.92e3T^{2} \)
97 \( 1 + 16.0T + 9.40e3T^{2} \)
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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

−11.25542750625243469257676578516, −9.865804184271321201583378095982, −9.332338757672133290949284011520, −8.060713492432782144432162606130, −6.85860696873900658279688325558, −5.51663598661810573293420350806, −4.81705893185234000399945979268, −3.64817410749758739240783256946, −1.60150975856027996411476577044, −0.76662869535222611198187164275, 1.94480993419065661402952168846, 3.64416542957247269723217275047, 5.21761122918855938607816785608, 6.26094251472231291609164363835, 6.62940295471663441784935166228, 7.56339429528171626325734094192, 8.474421273683446338029297676282, 10.26828130869401688566466024691, 10.83392520909760308518117880182, 11.34180030891588148576585829584

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