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Properties

Label 2-363-33.5-c2-0-40
Degree $2$
Conductor $363$
Sign $0.503 + 0.864i$
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  + (−2.73 + 0.889i)2-s + (−2.95 + 0.494i)3-s + (3.46 − 2.51i)4-s + (4.70 + 1.52i)5-s + (7.65 − 3.98i)6-s + (9.04 − 6.57i)7-s + (−0.474 + 0.653i)8-s + (8.51 − 2.92i)9-s − 14.2·10-s + (−9.00 + 9.15i)12-s + (−5.22 − 16.0i)13-s + (−18.9 + 26.0i)14-s + (−14.6 − 2.19i)15-s + (−4.57 + 14.0i)16-s + (0.0595 + 0.0193i)17-s + (−20.6 + 15.5i)18-s + ⋯
L(s)  = 1  + (−1.36 + 0.444i)2-s + (−0.986 + 0.164i)3-s + (0.865 − 0.629i)4-s + (0.941 + 0.305i)5-s + (1.27 − 0.664i)6-s + (1.29 − 0.938i)7-s + (−0.0593 + 0.0816i)8-s + (0.945 − 0.325i)9-s − 1.42·10-s + (−0.750 + 0.763i)12-s + (−0.401 − 1.23i)13-s + (−1.35 + 1.85i)14-s + (−0.978 − 0.146i)15-s + (−0.285 + 0.879i)16-s + (0.00350 + 0.00113i)17-s + (−1.14 + 0.865i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.503 + 0.864i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.547579 - 0.314791i\)
\(L(\frac12)\) \(\approx\) \(0.547579 - 0.314791i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.95 - 0.494i)T \)
11 \( 1 \)
good2 \( 1 + (2.73 - 0.889i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (-4.70 - 1.52i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (-9.04 + 6.57i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (5.22 + 16.0i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (-0.0595 - 0.0193i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (4.35 + 3.16i)T + (111. + 343. i)T^{2} \)
23 \( 1 - 8.69iT - 529T^{2} \)
29 \( 1 + (29.3 + 40.4i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (7.59 + 23.3i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-31.7 + 23.0i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (25.8 - 35.5i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 0.201T + 1.84e3T^{2} \)
47 \( 1 + (-8.76 + 12.0i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (93.1 - 30.2i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (8.51 + 11.7i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-9.06 + 27.8i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 82.8T + 4.48e3T^{2} \)
71 \( 1 + (25.3 + 8.25i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (11.3 - 8.21i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-15.5 - 47.7i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-57.2 - 18.6i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 40.3iT - 7.92e3T^{2} \)
97 \( 1 + (9.52 + 29.3i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−10.86261145796232209160949327307, −10.01655241865963247947204064049, −9.547789361698146497839914244683, −7.987008849704843451769198663817, −7.53927454049463283177272048828, −6.39290871615811394677215317743, −5.46637264658297102852930450837, −4.24665951575748721440408310776, −1.80630376564064128128906432460, −0.53559198882069974049752786965, 1.50646281420689383609255831773, 2.04232832771135769846658422665, 4.77805977319131977008210978644, 5.46203799614952081854106085723, 6.72432449335132546111259799489, 7.83570934643950869976551717981, 8.888247352202721192691086841393, 9.454093828319260433282554854312, 10.47338086135995064725893565541, 11.24409662770650175149895038555

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