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Properties

Label 2-363-33.8-c1-0-4
Degree $2$
Conductor $363$
Sign $-0.559 - 0.829i$
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  + (−1.93 + 1.40i)2-s + (−1.49 + 0.871i)3-s + (1.15 − 3.54i)4-s + (0.728 − 1.00i)5-s + (1.67 − 3.79i)6-s + (−2.68 − 0.874i)7-s + (1.28 + 3.94i)8-s + (1.48 − 2.60i)9-s + 2.96i·10-s + (1.36 + 6.31i)12-s + (1.96 + 2.70i)13-s + (6.44 − 2.09i)14-s + (−0.216 + 2.13i)15-s + (−1.99 − 1.44i)16-s + (3.35 + 2.43i)17-s + (0.800 + 7.13i)18-s + ⋯
L(s)  = 1  + (−1.36 + 0.995i)2-s + (−0.864 + 0.503i)3-s + (0.576 − 1.77i)4-s + (0.325 − 0.448i)5-s + (0.683 − 1.54i)6-s + (−1.01 − 0.330i)7-s + (0.453 + 1.39i)8-s + (0.493 − 0.869i)9-s + 0.938i·10-s + (0.394 + 1.82i)12-s + (0.545 + 0.750i)13-s + (1.72 − 0.559i)14-s + (−0.0560 + 0.551i)15-s + (−0.498 − 0.362i)16-s + (0.813 + 0.591i)17-s + (0.188 + 1.68i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.559 - 0.829i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.559 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190232 + 0.357745i\)
\(L(\frac12)\) \(\approx\) \(0.190232 + 0.357745i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 - 0.871i)T \)
11 \( 1 \)
good2 \( 1 + (1.93 - 1.40i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.728 + 1.00i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (2.68 + 0.874i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.96 - 2.70i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.35 - 2.43i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.32 - 0.756i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-1.67 + 5.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.158 - 0.115i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.23 - 9.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.21 - 6.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + (3.22 - 1.04i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.17 - 5.74i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-12.0 - 3.90i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.65 + 3.65i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (4.90 - 6.74i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.70 - 2.50i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.31 - 5.94i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.0 - 8.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.58iT - 89T^{2} \)
97 \( 1 + (1.83 - 1.33i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.39749385759010740380605457371, −10.31649597383768912248232305601, −9.840760478010172728076649608096, −9.095085190742367604883152055455, −8.128286844839280173229031046797, −6.78308177328922500685759284101, −6.32290922313553125931031602985, −5.35615073675253123093819385254, −3.84846844966238453704119550698, −1.14170913648077964555886486415, 0.58074782753289108729801184583, 2.22746430204921791331936569524, 3.36757723303028418412132940902, 5.47076068040655463851236545472, 6.53128419885707297949619648845, 7.44718723430537701131317019080, 8.528325990815092294685393881212, 9.530919153313480404713229513241, 10.39887973093064021349204255952, 10.79331962753832819765820006819

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