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Properties

Label 2-363-33.17-c1-0-13
Degree $2$
Conductor $363$
Sign $0.947 + 0.320i$
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  + (0.309 + 0.951i)2-s + (−1.64 + 0.556i)3-s + (0.809 − 0.587i)4-s + (−2.68 − 0.874i)5-s + (−1.03 − 1.38i)6-s + (−0.831 − 1.14i)7-s + (2.42 + 1.76i)8-s + (2.38 − 1.82i)9-s − 2.82i·10-s + (−1.00 + 1.41i)12-s + (4.03 − 1.31i)13-s + (0.831 − 1.14i)14-s + (4.89 − 0.0628i)15-s + (−0.309 + 0.951i)16-s + (1.85 − 5.70i)17-s + (2.47 + 1.70i)18-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.947 + 0.321i)3-s + (0.404 − 0.293i)4-s + (−1.20 − 0.390i)5-s + (−0.422 − 0.566i)6-s + (−0.314 − 0.432i)7-s + (0.858 + 0.623i)8-s + (0.793 − 0.608i)9-s − 0.894i·10-s + (−0.288 + 0.408i)12-s + (1.11 − 0.363i)13-s + (0.222 − 0.305i)14-s + (1.26 − 0.0162i)15-s + (−0.0772 + 0.237i)16-s + (0.449 − 1.38i)17-s + (0.582 + 0.400i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.947 + 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01544 - 0.166888i\)
\(L(\frac12)\) \(\approx\) \(1.01544 - 0.166888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.556i)T \)
11 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (2.68 + 0.874i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.831 + 1.14i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.03 + 1.31i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.85 + 5.70i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.49 + 3.43i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-1.61 + 1.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.618 + 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.47 - 4.70i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.85 + 3.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (1.66 - 2.28i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.37 + 1.74i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.65 + 9.15i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-9.41 - 3.05i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (2.68 + 0.874i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.831 + 1.14i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.03 + 1.31i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.94 - 15.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (0.618 + 1.90i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.47750276984836053464014639676, −10.67854619941979229913353974459, −9.698581685720283540644424733222, −8.322341569044083054348201288953, −7.30076354486451141897348100271, −6.66295392279903392750198645906, −5.47259339432102862653389124728, −4.68928019924797214950462724040, −3.49407391937860238627529053898, −0.812133292760681154756368517231, 1.57719128515992461880181746263, 3.39141934965211634748696968432, 4.10805070480544499502035427517, 5.73615152481616177678311052552, 6.71984093201842392430928651860, 7.57848827846414059746700638553, 8.493531690857645651755721245422, 10.22616663103042775757256267446, 10.81614160344452942178966070502, 11.61384305544024814784729129631

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