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Properties

Label 2-363-11.4-c1-0-17
Degree $2$
Conductor $363$
Sign $0.0694 - 0.997i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 1.31i)2-s + (0.309 − 0.951i)3-s + (0.927 + 2.85i)4-s + (−1.61 + 1.17i)5-s + (−1.80 + 1.31i)6-s + (−1.38 − 4.25i)7-s + (0.690 − 2.12i)8-s + (−0.809 − 0.587i)9-s + 4.47·10-s + 2.99·12-s + (−3.09 + 9.51i)14-s + (0.618 + 1.90i)15-s + (0.809 − 0.587i)16-s + (−3.61 + 2.62i)17-s + (0.690 + 2.12i)18-s + (−1.38 + 4.25i)19-s + ⋯
L(s)  = 1  + (−1.27 − 0.929i)2-s + (0.178 − 0.549i)3-s + (0.463 + 1.42i)4-s + (−0.723 + 0.525i)5-s + (−0.738 + 0.536i)6-s + (−0.522 − 1.60i)7-s + (0.244 − 0.751i)8-s + (−0.269 − 0.195i)9-s + 1.41·10-s + 0.866·12-s + (−0.825 + 2.54i)14-s + (0.159 + 0.491i)15-s + (0.202 − 0.146i)16-s + (−0.877 + 0.637i)17-s + (0.162 + 0.501i)18-s + (−0.317 + 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.31i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.61 - 1.17i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.38 + 4.25i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.61 - 2.62i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.38 - 4.25i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.618 - 1.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.38 - 4.25i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
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 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.23 - 5.25i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (-6.47 + 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.76 + 8.50i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.8 + 7.88i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.23 + 5.25i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
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

−10.57406753056479389385192318086, −10.11311115078579586624472779476, −8.911596981415651533460826170864, −7.919997173281999995308655437635, −7.36742244047383441975759703259, −6.37427797255033564209703097513, −4.06724442962360687649635614436, −3.18757331967181542271751575697, −1.59237093743335086277336395191, 0, 2.57095219232293235532905053543, 4.34907605974048514928284053301, 5.63493262137862259144867806582, 6.53554551506523268766400769498, 7.76125293115082883441175143672, 8.619710990171056090780648214955, 9.095155075075594348895604095924, 9.769634080470012183104372079288, 11.00816007690232061038191523261

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