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Properties

Label 2-363-33.2-c1-0-20
Degree $2$
Conductor $363$
Sign $0.231 + 0.972i$
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  + (−0.756 + 2.32i)2-s + (0.704 − 1.58i)3-s + (−3.23 − 2.35i)4-s + (−1.64 + 0.535i)5-s + (3.15 + 2.83i)6-s + (−0.831 + 1.14i)7-s + (3.96 − 2.87i)8-s + (−2.00 − 2.22i)9-s − 4.24i·10-s + (−5.99 + 3.46i)12-s + (−2.68 − 0.874i)13-s + (−2.03 − 2.80i)14-s + (−0.313 + 2.98i)15-s + (1.23 + 3.80i)16-s + (−0.756 − 2.32i)17-s + (6.71 − 2.98i)18-s + ⋯
L(s)  = 1  + (−0.535 + 1.64i)2-s + (0.406 − 0.913i)3-s + (−1.61 − 1.17i)4-s + (−0.736 + 0.239i)5-s + (1.28 + 1.15i)6-s + (−0.314 + 0.432i)7-s + (1.40 − 1.01i)8-s + (−0.669 − 0.743i)9-s − 1.34i·10-s + (−1.73 + 0.999i)12-s + (−0.746 − 0.242i)13-s + (−0.544 − 0.749i)14-s + (−0.0809 + 0.770i)15-s + (0.309 + 0.951i)16-s + (−0.183 − 0.565i)17-s + (1.58 − 0.704i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203990 - 0.161096i\)
\(L(\frac12)\) \(\approx\) \(0.203990 - 0.161096i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.704 + 1.58i)T \)
11 \( 1 \)
good2 \( 1 + (0.756 - 2.32i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.64 - 0.535i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.831 - 1.14i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.68 + 0.874i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.756 + 2.32i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.66 + 2.28i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.66iT - 23T^{2} \)
29 \( 1 + (3.96 + 2.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.927 + 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.42 + 1.76i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.92 - 5.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.89iT - 43T^{2} \)
47 \( 1 + (-4.07 - 5.60i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.29 - 1.07i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.01 + 1.40i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.68 + 0.874i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (1.64 - 0.535i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.32 + 4.57i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.41 + 3.05i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.756 - 2.32i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.66iT - 89T^{2} \)
97 \( 1 + (-4.01 + 12.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.31331477832288996645420218504, −9.801593377665983609034050716062, −8.950158316199723359890676341347, −8.151201110418004787625318716219, −7.45367849704323360663068990233, −6.71431764253646266481691000515, −5.89274773820463639692409435495, −4.55045045914802862718863411345, −2.72634854027493517537561991765, −0.19274590733159826339110527547, 2.02307094757666301680032869944, 3.57141319239467591034409213708, 3.91115798063216536973640757251, 5.23174953969505999453358241808, 7.34845686968204021952018259356, 8.431800872352040807775219925795, 9.090677388279423773268545865259, 10.10747646040772974411431930008, 10.45940843460763552704459136310, 11.58397129237667257227058967140

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