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Properties

Label 2-363-33.8-c1-0-24
Degree $2$
Conductor $363$
Sign $0.166 + 0.986i$
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 + (1.72 − 0.175i)3-s + (1.15 − 3.54i)4-s + (−0.728 + 1.00i)5-s + (3.09 − 2.76i)6-s + (−2.68 − 0.874i)7-s + (−1.28 − 3.94i)8-s + (2.93 − 0.603i)9-s + 2.96i·10-s + (1.36 − 6.31i)12-s + (1.96 + 2.70i)13-s + (−6.44 + 2.09i)14-s + (−1.07 + 1.85i)15-s + (−1.99 − 1.44i)16-s + (−3.35 − 2.43i)17-s + (4.84 − 5.30i)18-s + ⋯
L(s)  = 1  + (1.36 − 0.995i)2-s + (0.994 − 0.101i)3-s + (0.576 − 1.77i)4-s + (−0.325 + 0.448i)5-s + (1.26 − 1.12i)6-s + (−1.01 − 0.330i)7-s + (−0.453 − 1.39i)8-s + (0.979 − 0.201i)9-s + 0.938i·10-s + (0.394 − 1.82i)12-s + (0.545 + 0.750i)13-s + (−1.72 + 0.559i)14-s + (−0.278 + 0.479i)15-s + (−0.498 − 0.362i)16-s + (−0.813 − 0.591i)17-s + (1.14 − 1.25i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.166 + 0.986i$
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: \((2,\ 363,\ (\ :1/2),\ 0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45705 - 2.07652i\)
\(L(\frac12)\) \(\approx\) \(2.45705 - 2.07652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.175i)T \)
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

−11.33346004435748366159033743036, −10.55217694370633370967159632802, −9.627294271223663925144224042395, −8.636868139435945530832905987761, −7.08122920765559630188723833966, −6.43617165302227541185655348253, −4.81158054440556071159563977253, −3.70534181075039688135134034360, −3.18663351080895905103974551262, −1.89869903893622004170126994527, 2.69657096337793610169465988939, 3.76124101229989286237758967073, 4.52630909905839247999717686930, 5.88853046071593627555238412560, 6.66889639470100054718996054259, 7.79534150083665078509261432965, 8.524021716550344776091540606565, 9.515409406495565407935153590453, 10.78256439540639705933592767455, 12.31819781185110068950897313687

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