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Properties

Label 2-363-11.3-c1-0-8
Degree $2$
Conductor $363$
Sign $0.995 + 0.0913i$
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  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (1.61 + 1.17i)5-s + (0.809 + 0.587i)6-s + (1.23 − 3.80i)7-s + (−0.927 − 2.85i)8-s + (−0.809 + 0.587i)9-s − 2·10-s + 0.999·12-s + (1.61 − 1.17i)13-s + (1.23 + 3.80i)14-s + (0.618 − 1.90i)15-s + (0.809 + 0.587i)16-s + (1.61 + 1.17i)17-s + (0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (−0.154 + 0.475i)4-s + (0.723 + 0.525i)5-s + (0.330 + 0.239i)6-s + (0.467 − 1.43i)7-s + (−0.327 − 1.00i)8-s + (−0.269 + 0.195i)9-s − 0.632·10-s + 0.288·12-s + (0.448 − 0.326i)13-s + (0.330 + 1.01i)14-s + (0.159 − 0.491i)15-s + (0.202 + 0.146i)16-s + (0.392 + 0.285i)17-s + (0.0728 − 0.224i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.995 + 0.0913i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.995 + 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05846 - 0.0484686i\)
\(L(\frac12)\) \(\approx\) \(1.05846 - 0.0484686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.23 + 3.80i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.61 + 1.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.61 - 1.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.47 + 4.70i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.85 + 5.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.618 + 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-2.47 - 7.60i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.85 - 3.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.23 - 3.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.85 + 3.52i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.32 - 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.70 + 7.05i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (1.61 - 1.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.14035074097355378837046640489, −10.51848162273505856752290011408, −9.532947179135651459871728145427, −8.428035839641687917853336516087, −7.51155274978116669969181040823, −6.95942308197040727936427706270, −5.94311890327935492725520166309, −4.37718058628109478588777029219, −3.05472149613818657601559610566, −1.08779967835696234453711612737, 1.45331848047328978567482974789, 2.76978359856214426281299607856, 4.84801067984529428985842469974, 5.41176829238472466554664166835, 6.31574932757034554537252582885, 8.246268470036361314252354269464, 9.012676679030885163482067928499, 9.460846099457401586346010017683, 10.38832560941731692992475797034, 11.40856045521685175147980469806

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