↑

Properties

Label 2-585-13.9-c1-0-19
Degree $2$
Conductor $585$
Sign $0.0128 + 0.999i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  + (−0.309 + 0.535i)2-s + (0.809 + 1.40i)4-s − 5-s + (−2.11 − 3.66i)7-s − 2.23·8-s + (0.309 − 0.535i)10-s + (−0.118 + 0.204i)11-s + (−1 − 3.46i)13-s + 2.61·14-s + (−0.927 + 1.60i)16-s + (−1.73 − 3.00i)17-s + (−2.11 − 3.66i)19-s + (−0.809 − 1.40i)20-s + (−0.0729 − 0.126i)22-s + (1.88 − 3.25i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.378i)2-s + (0.404 + 0.700i)4-s − 0.447·5-s + (−0.800 − 1.38i)7-s − 0.790·8-s + (0.0977 − 0.169i)10-s + (−0.0355 + 0.0616i)11-s + (−0.277 − 0.960i)13-s + 0.699·14-s + (−0.231 + 0.401i)16-s + (−0.421 − 0.729i)17-s + (−0.485 − 0.841i)19-s + (−0.180 − 0.313i)20-s + (−0.0155 − 0.0269i)22-s + (0.392 − 0.679i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437999 - 0.432418i\)
\(L(\frac12)\) \(\approx\) \(0.437999 - 0.432418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + (0.309 - 0.535i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.118 - 0.204i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.73 + 3.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.11 + 3.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.88 + 3.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.73 - 6.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.97 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.11 - 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (0.354 + 0.613i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.20 + 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.35 + 2.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.11 + 5.40i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.73 - 3.00i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.70323563179555722153570479996, −9.490267378032233923526017124576, −8.634127983199081182706011819775, −7.48672294730012299086966117904, −7.18577817949403755617788900360, −6.28459494911897342099917778355, −4.75138525560885738486016018059, −3.64664699179982748449025053662, −2.78320810426346211461340491440, −0.34666713397702527086615944668, 1.85338986899159605204225231246, 2.85824117381228768039215392628, 4.22578025241589651508972337880, 5.73971289702522364990185617485, 6.15771796462568575908585185226, 7.27855243821300389808197538421, 8.588809140134311489226908721013, 9.279759827797516489549651628550, 9.951864488539769568573126607849, 11.00735536856705118305422813292

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