↑

Properties

Label 2-363-11.4-c1-0-13
Degree $2$
Conductor $363$
Sign $0.927 + 0.374i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 + 0.951i)2-s + (0.309 − 0.951i)3-s + (0.190 + 0.587i)4-s + (0.309 − 0.224i)5-s + (1.30 − 0.951i)6-s + (−0.927 − 2.85i)7-s + (0.690 − 2.12i)8-s + (−0.809 − 0.587i)9-s + 0.618·10-s + 0.618·12-s + (5.04 + 3.66i)13-s + (1.5 − 4.61i)14-s + (−0.118 − 0.363i)15-s + (3.92 − 2.85i)16-s + (−0.5 + 0.363i)17-s + (−0.499 − 1.53i)18-s + ⋯
L(s)  = 1  + (0.925 + 0.672i)2-s + (0.178 − 0.549i)3-s + (0.0954 + 0.293i)4-s + (0.138 − 0.100i)5-s + (0.534 − 0.388i)6-s + (−0.350 − 1.07i)7-s + (0.244 − 0.751i)8-s + (−0.269 − 0.195i)9-s + 0.195·10-s + 0.178·12-s + (1.39 + 1.01i)13-s + (0.400 − 1.23i)14-s + (−0.0304 − 0.0937i)15-s + (0.981 − 0.713i)16-s + (−0.121 + 0.0881i)17-s + (−0.117 − 0.362i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.927 + 0.374i$
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: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18789 - 0.424966i\)
\(L(\frac12)\) \(\approx\) \(2.18789 - 0.424966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 0.951i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.309 + 0.224i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.927 + 2.85i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.04 - 3.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.263 + 0.812i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.11 - 2.26i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.30 + 4.02i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.83 - 5.65i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + (0.190 - 0.587i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.97 - 4.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.64 + 5.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.927 + 0.673i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + (11.7 - 8.55i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.381 + 1.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.427 - 0.310i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.2 - 7.46i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 9.47T + 89T^{2} \)
97 \( 1 + (12.1 + 8.83i)T + (29.9 + 92.2i)T^{2} \)
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

−11.53153967875107988974687381461, −10.50175440600152185558390426096, −9.502511511838861163313282369242, −8.359442629740444964680472338572, −7.16174617959020833457234934033, −6.59499183901965007070164906760, −5.69529136386931716985065287513, −4.32439820119788798275205486366, −3.53235334792978867159417635623, −1.35730264739982931749737324215, 2.29116944987890974546669924597, 3.28885919867012159462202789842, 4.24391109747656867894977990437, 5.53715165773940391836273667989, 6.13425150205134231176233639154, 8.101826672997573657269312259242, 8.629188344538009554208504873035, 9.925672497820429654364093572375, 10.67292136303796175608522307314, 11.73150049139990531066128980259

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