↑

Properties

Label 2-285-285.197-c1-0-26
Degree $2$
Conductor $285$
Sign $-0.0810 + 0.996i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + (1.01 + 0.272i)2-s + (−1.49 + 0.872i)3-s + (−0.771 − 0.445i)4-s + (−1.26 − 1.84i)5-s + (−1.75 + 0.480i)6-s + (1.17 + 1.17i)7-s + (−2.15 − 2.15i)8-s + (1.47 − 2.61i)9-s + (−0.787 − 2.21i)10-s − 3.18i·11-s + (1.54 − 0.00697i)12-s + (−1.15 − 4.32i)13-s + (0.874 + 1.51i)14-s + (3.50 + 1.64i)15-s + (−0.712 − 1.23i)16-s + (−3.91 − 1.04i)17-s + ⋯
L(s)  = 1  + (0.719 + 0.192i)2-s + (−0.863 + 0.503i)3-s + (−0.385 − 0.222i)4-s + (−0.566 − 0.823i)5-s + (−0.718 + 0.195i)6-s + (0.443 + 0.443i)7-s + (−0.761 − 0.761i)8-s + (0.492 − 0.870i)9-s + (−0.249 − 0.701i)10-s − 0.960i·11-s + (0.445 − 0.00201i)12-s + (−0.321 − 1.19i)13-s + (0.233 + 0.404i)14-s + (0.904 + 0.425i)15-s + (−0.178 − 0.308i)16-s + (−0.950 − 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0810 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0810 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.0810 + 0.996i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.0810 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541975 - 0.587854i\)
\(L(\frac12)\) \(\approx\) \(0.541975 - 0.587854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 - 0.872i)T \)
5 \( 1 + (1.26 + 1.84i)T \)
19 \( 1 + (-4.35 + 0.0784i)T \)
good2 \( 1 + (-1.01 - 0.272i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-1.17 - 1.17i)T + 7iT^{2} \)
11 \( 1 + 3.18iT - 11T^{2} \)
13 \( 1 + (1.15 + 4.32i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.91 + 1.04i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (5.98 - 1.60i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.01 + 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.68T + 31T^{2} \)
37 \( 1 + (-0.612 - 0.612i)T + 37iT^{2} \)
41 \( 1 + (-4.82 + 2.78i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.4 - 2.79i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.290 + 1.08i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.0565 - 0.0151i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.45 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.26 - 9.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.21 + 2.20i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.39 + 1.38i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.6 + 3.64i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.2 + 7.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.32 - 6.32i)T + 83iT^{2} \)
89 \( 1 + (-1.42 + 2.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.574 - 2.14i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.76104207886845886000705644060, −10.85887557740848386068050877979, −9.647416160518219283345210655601, −8.853855615451764237519548047582, −7.69050509707402569418011101083, −6.01677100499283201167432844451, −5.41352523422234004282561135112, −4.59070080692944374502043168733, −3.52740767174313327672897390065, −0.53889623785307160164398514341, 2.24578390716939542421556049944, 4.08752044077522637164798127576, 4.67034494178998022240893985156, 6.08371534344208689459467328815, 7.15344807484397318084958316029, 7.84200933488094420696132910089, 9.337901235530192378058449215403, 10.60115684733065942689443868581, 11.43083929235309360378379697556, 12.06117477759578580838983812978

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