↑

Properties

Label 2-585-13.8-c2-0-31
Degree $2$
Conductor $585$
Sign $0.783 - 0.621i$
Analytic cond. $15.9400$
Root an. cond. $3.99250$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.10 + 2.10i)2-s + 4.82i·4-s + (−1.58 − 1.58i)5-s + (3.75 − 3.75i)7-s + (−1.74 + 1.74i)8-s − 6.64i·10-s + (10.5 − 10.5i)11-s + (2.32 + 12.7i)13-s + 15.7·14-s + 11.9·16-s − 21.6i·17-s + (−5.89 − 5.89i)19-s + (7.63 − 7.63i)20-s + 44.3·22-s + 4.11i·23-s + ⋯
L(s)  = 1  + (1.05 + 1.05i)2-s + 1.20i·4-s + (−0.316 − 0.316i)5-s + (0.536 − 0.536i)7-s + (−0.217 + 0.217i)8-s − 0.664i·10-s + (0.960 − 0.960i)11-s + (0.178 + 0.983i)13-s + 1.12·14-s + 0.749·16-s − 1.27i·17-s + (−0.310 − 0.310i)19-s + (0.381 − 0.381i)20-s + 2.01·22-s + 0.179i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.783 - 0.621i$
Analytic conductor: \(15.9400\)
Root analytic conductor: \(3.99250\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1),\ 0.783 - 0.621i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.430649133\)
\(L(\frac12)\) \(\approx\) \(3.430649133\)
\(L(2)\) 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 \( 1 + (1.58 + 1.58i)T \)
13 \( 1 + (-2.32 - 12.7i)T \)
good2 \( 1 + (-2.10 - 2.10i)T + 4iT^{2} \)
7 \( 1 + (-3.75 + 3.75i)T - 49iT^{2} \)
11 \( 1 + (-10.5 + 10.5i)T - 121iT^{2} \)
17 \( 1 + 21.6iT - 289T^{2} \)
19 \( 1 + (5.89 + 5.89i)T + 361iT^{2} \)
23 \( 1 - 4.11iT - 529T^{2} \)
29 \( 1 - 29.2T + 841T^{2} \)
31 \( 1 + (-15.1 - 15.1i)T + 961iT^{2} \)
37 \( 1 + (-3.97 + 3.97i)T - 1.36e3iT^{2} \)
41 \( 1 + (14.4 + 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 - 79.6iT - 1.84e3T^{2} \)
47 \( 1 + (-27.0 + 27.0i)T - 2.20e3iT^{2} \)
53 \( 1 + 32.4T + 2.80e3T^{2} \)
59 \( 1 + (6.91 - 6.91i)T - 3.48e3iT^{2} \)
61 \( 1 + 56.7T + 3.72e3T^{2} \)
67 \( 1 + (78.6 + 78.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (-41.8 - 41.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (-89.5 + 89.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 21.9T + 6.24e3T^{2} \)
83 \( 1 + (35.5 + 35.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (26.2 - 26.2i)T - 7.92e3iT^{2} \)
97 \( 1 + (50.1 + 50.1i)T + 9.40e3iT^{2} \)
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

−10.85494543408423409023129703967, −9.479116521195518131526485231079, −8.585480145660544753661451436507, −7.66239917982851361025163664526, −6.79266360688036792035933452215, −6.12529410865711796468246517006, −4.84518547990477891834311441424, −4.35776573391892109656472937642, −3.25297581629899591934762101550, −1.11892966994138880613552437799, 1.47221187732409938096205909304, 2.54193264170755372639152339634, 3.73167411172834985863131348863, 4.46469081756528279098265657728, 5.52618410897242482229313515190, 6.50255110544761960851400678640, 7.83653828863965248620756861639, 8.629887244117751024155481784228, 10.00284543149018697782263593778, 10.59778699827319049364150196340

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