↑

Properties

Label 2-363-11.3-c1-0-10
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

Related objects

Downloads

Learn more

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: $\chi_{363} (124, \cdot )$
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 \( 1 + (1.61 - 1.17i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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