↑

Properties

Label 2-363-11.3-c1-0-13
Degree $2$
Conductor $363$
Sign $0.659 + 0.751i$
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  + (2.04 − 1.48i)2-s + (0.309 + 0.951i)3-s + (1.35 − 4.15i)4-s + (1.91 + 1.39i)5-s + (2.04 + 1.48i)6-s + (−0.244 + 0.753i)7-s + (−1.85 − 5.69i)8-s + (−0.809 + 0.587i)9-s + 5.98·10-s + 4.37·12-s + (−3.32 + 2.41i)13-s + (0.618 + 1.90i)14-s + (−0.733 + 2.25i)15-s + (−5.15 − 3.74i)16-s + (−4.84 − 3.51i)17-s + (−0.780 + 2.40i)18-s + ⋯
L(s)  = 1  + (1.44 − 1.04i)2-s + (0.178 + 0.549i)3-s + (0.675 − 2.07i)4-s + (0.858 + 0.623i)5-s + (0.833 + 0.605i)6-s + (−0.0925 + 0.284i)7-s + (−0.654 − 2.01i)8-s + (−0.269 + 0.195i)9-s + 1.89·10-s + 1.26·12-s + (−0.921 + 0.669i)13-s + (0.165 + 0.508i)14-s + (−0.189 + 0.582i)15-s + (−1.28 − 0.936i)16-s + (−1.17 − 0.853i)17-s + (−0.183 + 0.565i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.659 + 0.751i$
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: \((2,\ 363,\ (\ :1/2),\ 0.659 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.84968 - 1.29114i\)
\(L(\frac12)\) \(\approx\) \(2.84968 - 1.29114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 1.48i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.91 - 1.39i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.244 - 0.753i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.32 - 2.41i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.84 + 3.51i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.31 + 4.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (-0.780 + 2.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.96 - 4.33i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.54 + 4.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.75 - 5.41i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.63T + 43T^{2} \)
47 \( 1 + (0.387 + 1.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.6 - 7.70i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.16 + 1.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 + (0.602 + 0.437i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.29 - 7.06i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.72 + 3.43i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.88 - 5.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.37T + 89T^{2} \)
97 \( 1 + (-10.1 + 7.34i)T + (29.9 - 92.2i)T^{2} \)
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

−11.18767086752607706112717163110, −10.80339735622160562169578449168, −9.676402940056464003672878268581, −9.142831784581604241495592782077, −7.05433935748048453147273445344, −6.11467717295028886089627366000, −5.03959251819577741238894872750, −4.27922432912401501028706396725, −2.80196072639219902061330052493, −2.25003971314852961356507237130, 2.17223613167512368719926516358, 3.68043327897373809642504144276, 4.88395399425372210510049681943, 5.73585399959510680874266753330, 6.53483924294312865866600443418, 7.50090382129395780465298103434, 8.363472265141825207909613869262, 9.507086387558719661938609624344, 10.84771112839467234918442898736, 12.19288765028738570115195504986

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