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Properties

Label 2-363-11.3-c1-0-4
Degree $2$
Conductor $363$
Sign $0.782 - 0.622i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.782 - 0.622i$
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: \((2,\ 363,\ (\ :1/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56244 + 0.545927i\)
\(L(\frac12)\) \(\approx\) \(1.56244 + 0.545927i\)
\(L(\frac{3}{2})\) 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

−11.77031720719592189043859172571, −10.95081130118889611654642108973, −9.596710929943466167605038587549, −8.857557702407961576245528068263, −7.73087969932428976566360110184, −6.52641738903461047931731754318, −5.72379071659773150020689052389, −4.59680674801099632065148050062, −2.82018581473423530918529981518, −2.36026984055612069969970144140, 1.04179043386401459710073506741, 3.41392332888437310674738873242, 4.64689388108015808252163385028, 5.21566021857320539991304636464, 6.45536507269735971089444455845, 7.14275722682849799658408568502, 8.741389982839601996509955212134, 9.751132638106307679571016945129, 10.26096586068518123325551155818, 11.04374390321835708829616362653

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