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Properties

Label 2-363-33.8-c1-0-0
Degree $2$
Conductor $363$
Sign $0.262 + 0.965i$
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  + (−1.53 + 1.11i)2-s + (−0.0877 + 1.72i)3-s + (0.499 − 1.53i)4-s + (−1.53 + 2.11i)5-s + (−1.79 − 2.76i)6-s + (−0.690 − 0.224i)7-s + (−0.224 − 0.690i)8-s + (−2.98 − 0.303i)9-s − 4.97i·10-s + (2.61 + 0.999i)12-s + (−1.80 − 2.48i)13-s + (1.31 − 0.427i)14-s + (−3.52 − 2.84i)15-s + (3.73 + 2.71i)16-s + (2.12 + 1.54i)17-s + (4.93 − 2.86i)18-s + ⋯
L(s)  = 1  + (−1.08 + 0.790i)2-s + (−0.0506 + 0.998i)3-s + (0.249 − 0.769i)4-s + (−0.688 + 0.947i)5-s + (−0.734 − 1.12i)6-s + (−0.261 − 0.0848i)7-s + (−0.0793 − 0.244i)8-s + (−0.994 − 0.101i)9-s − 1.57i·10-s + (0.755 + 0.288i)12-s + (−0.501 − 0.690i)13-s + (0.351 − 0.114i)14-s + (−0.911 − 0.735i)15-s + (0.934 + 0.678i)16-s + (0.515 + 0.374i)17-s + (1.16 − 0.676i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.262 + 0.965i$
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: \((2,\ 363,\ (\ :1/2),\ 0.262 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0985499 - 0.0753508i\)
\(L(\frac12)\) \(\approx\) \(0.0985499 - 0.0753508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0877 - 1.72i)T \)
11 \( 1 \)
good2 \( 1 + (1.53 - 1.11i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.53 - 2.11i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.690 + 0.224i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.80 + 2.48i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.12 - 1.54i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.04 - 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.76iT - 23T^{2} \)
29 \( 1 + (1.17 - 3.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.690 + 0.502i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.951 + 2.92i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.62iT - 43T^{2} \)
47 \( 1 + (-6.96 + 2.26i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.85 + 3.92i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.48 + 0.809i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.5 - 3.44i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + (6.06 - 8.35i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (14.4 + 4.70i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.28 - 8.64i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.7 + 8.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.47iT - 89T^{2} \)
97 \( 1 + (-5.35 + 3.88i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.87156382250105123039339271321, −10.60147638143059353142231530272, −10.39426423420019274582175091706, −9.341698283769480497977202284292, −8.415589487887634532034496039437, −7.62106152262599747257339271452, −6.67999834896896446184228091651, −5.64178627291174393102344012215, −4.05381155804290206205801057155, −3.07030284080205704574213372561, 0.12303124040700813944607700699, 1.50003477949630856131553217432, 2.82549936478896593049525913961, 4.60108521860180637569954720900, 5.95702369684521557288746544291, 7.30753092606337024442834586350, 8.108025778291836055191858379208, 8.859521867224601401676667145833, 9.591158153695011039925553037970, 10.81282012653317009554326258146

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