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Properties

Label 2-363-3.2-c2-0-18
Degree $2$
Conductor $363$
Sign $0.581 + 0.813i$
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  − 3.83i·2-s + (1.74 + 2.44i)3-s − 10.6·4-s + 0.620i·5-s + (9.35 − 6.68i)6-s + 1.92·7-s + 25.6i·8-s + (−2.92 + 8.51i)9-s + 2.37·10-s + (−18.6 − 26.1i)12-s + 15.9·13-s − 7.38i·14-s + (−1.51 + 1.08i)15-s + 55.5·16-s + 14.4i·17-s + (32.6 + 11.1i)18-s + ⋯
L(s)  = 1  − 1.91i·2-s + (0.581 + 0.813i)3-s − 2.67·4-s + 0.124i·5-s + (1.55 − 1.11i)6-s + 0.275·7-s + 3.20i·8-s + (−0.324 + 0.945i)9-s + 0.237·10-s + (−1.55 − 2.17i)12-s + 1.22·13-s − 0.527i·14-s + (−0.101 + 0.0721i)15-s + 3.47·16-s + 0.849i·17-s + (1.81 + 0.622i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.581 + 0.813i$
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: \((2,\ 363,\ (\ :1),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56660 - 0.806313i\)
\(L(\frac12)\) \(\approx\) \(1.56660 - 0.806313i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.74 - 2.44i)T \)
11 \( 1 \)
good2 \( 1 + 3.83iT - 4T^{2} \)
5 \( 1 - 0.620iT - 25T^{2} \)
7 \( 1 - 1.92T + 49T^{2} \)
13 \( 1 - 15.9T + 169T^{2} \)
17 \( 1 - 14.4iT - 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 - 10.6iT - 529T^{2} \)
29 \( 1 + 19.1iT - 841T^{2} \)
31 \( 1 - 22.0T + 961T^{2} \)
37 \( 1 + 20.6T + 1.36e3T^{2} \)
41 \( 1 + 27.0iT - 1.68e3T^{2} \)
43 \( 1 - 53.1T + 1.84e3T^{2} \)
47 \( 1 - 45.7iT - 2.20e3T^{2} \)
53 \( 1 - 60.4iT - 2.80e3T^{2} \)
59 \( 1 - 40.5iT - 3.48e3T^{2} \)
61 \( 1 - 64.1T + 3.72e3T^{2} \)
67 \( 1 + 25.7T + 4.48e3T^{2} \)
71 \( 1 + 114. iT - 5.04e3T^{2} \)
73 \( 1 + 84.4T + 5.32e3T^{2} \)
79 \( 1 + 65.2T + 6.24e3T^{2} \)
83 \( 1 - 89.6iT - 6.88e3T^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 - 62.3T + 9.40e3T^{2} \)
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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

−10.90100782013522246878646218680, −10.42762033452153118984937416793, −9.444942733790334065330107551628, −8.762486826631988781178445295883, −7.938533648430953360370694163265, −5.67916483225806509784547722034, −4.50709250235903774567071930369, −3.67416689750759673964596851824, −2.75921139075967615633774935140, −1.36899729105193982361131991211, 0.922585019642760219447578344224, 3.38200892655922521586790846395, 4.77249962642509766971553123020, 5.84722270602898158183758848660, 6.76963098106318144503195735825, 7.43002628470310141640753091699, 8.464215712566780599656112947882, 8.812626361330991931320688559287, 9.909750515249672007068543717006, 11.55312653441264245389204918652

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