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Properties

Label 2-363-33.29-c1-0-27
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: $\chi_{363} (161, \cdot )$
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

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

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