↑

Properties

Label 2-585-5.4-c3-0-41
Degree $2$
Conductor $585$
Sign $0.874 - 0.484i$
Analytic cond. $34.5161$
Root an. cond. $5.87504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.66i·2-s + 5.21·4-s + (−5.41 − 9.77i)5-s − 1.83i·7-s + 22.0i·8-s + (16.3 − 9.03i)10-s − 40.1·11-s + 13i·13-s + 3.06·14-s + 4.96·16-s − 31.2i·17-s + 150.·19-s + (−28.2 − 51.0i)20-s − 66.9i·22-s + 169. i·23-s + ⋯
L(s)  = 1  + 0.589i·2-s + 0.652·4-s + (−0.484 − 0.874i)5-s − 0.0990i·7-s + 0.974i·8-s + (0.515 − 0.285i)10-s − 1.10·11-s + 0.277i·13-s + 0.0584·14-s + 0.0776·16-s − 0.446i·17-s + 1.81·19-s + (−0.316 − 0.570i)20-s − 0.649i·22-s + 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(34.5161\)
Root analytic conductor: \(5.87504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :3/2),\ 0.874 - 0.484i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.165583020\)
\(L(\frac12)\) \(\approx\) \(2.165583020\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.41 + 9.77i)T \)
13 \( 1 - 13iT \)
good2 \( 1 - 1.66iT - 8T^{2} \)
7 \( 1 + 1.83iT - 343T^{2} \)
11 \( 1 + 40.1T + 1.33e3T^{2} \)
17 \( 1 + 31.2iT - 4.91e3T^{2} \)
19 \( 1 - 150.T + 6.85e3T^{2} \)
23 \( 1 - 169. iT - 1.21e4T^{2} \)
29 \( 1 - 151.T + 2.43e4T^{2} \)
31 \( 1 - 71.3T + 2.97e4T^{2} \)
37 \( 1 + 298. iT - 5.06e4T^{2} \)
41 \( 1 - 360.T + 6.89e4T^{2} \)
43 \( 1 + 186. iT - 7.95e4T^{2} \)
47 \( 1 + 304. iT - 1.03e5T^{2} \)
53 \( 1 + 590. iT - 1.48e5T^{2} \)
59 \( 1 - 767.T + 2.05e5T^{2} \)
61 \( 1 - 714.T + 2.26e5T^{2} \)
67 \( 1 - 35.1iT - 3.00e5T^{2} \)
71 \( 1 + 148.T + 3.57e5T^{2} \)
73 \( 1 - 658. iT - 3.89e5T^{2} \)
79 \( 1 + 752.T + 4.93e5T^{2} \)
83 \( 1 + 123. iT - 5.71e5T^{2} \)
89 \( 1 - 949.T + 7.04e5T^{2} \)
97 \( 1 - 131. iT - 9.12e5T^{2} \)
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

−10.33363783819220409487835120432, −9.387032525842773239462818954784, −8.356483082459834381986881000906, −7.58488990760245401802142979308, −7.06592103145680511098288875502, −5.49378527552180220450774408028, −5.24507049118014730520772490551, −3.71153084965557858550563403189, −2.41607546236648611883325053111, −0.899824697064044058668305631205, 0.876111830196004181709016124283, 2.61447327046754849317185442292, 3.00788490908907341008362986058, 4.35520785460276214371120170528, 5.75018286730445303337815282613, 6.70598235382101088807415130976, 7.55010596421259909627789332116, 8.269146148721374406377632591699, 9.783528443670521704577362316787, 10.41057898089335889819794805311

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