↑

Properties

Label 2-585-13.12-c1-0-2
Degree $2$
Conductor $585$
Sign $-0.953 + 0.301i$
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  + 2.08i·2-s − 2.35·4-s + i·5-s − 1.35i·7-s − 0.734i·8-s − 2.08·10-s + 3.73i·11-s + (1.08 + 3.43i)13-s + 2.82·14-s − 3.17·16-s − 2.70·17-s − 0.438i·19-s − 2.35i·20-s − 7.79·22-s − 5.08·23-s + ⋯
L(s)  = 1  + 1.47i·2-s − 1.17·4-s + 0.447i·5-s − 0.510i·7-s − 0.259i·8-s − 0.659·10-s + 1.12i·11-s + (0.301 + 0.953i)13-s + 0.753·14-s − 0.793·16-s − 0.655·17-s − 0.100i·19-s − 0.525i·20-s − 1.66·22-s − 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.953 + 0.301i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.953 + 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.173324 - 1.12401i\)
\(L(\frac12)\) \(\approx\) \(0.173324 - 1.12401i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-1.08 - 3.43i)T \)
good2 \( 1 - 2.08iT - 2T^{2} \)
7 \( 1 + 1.35iT - 7T^{2} \)
11 \( 1 - 3.73iT - 11T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + 0.438iT - 19T^{2} \)
23 \( 1 + 5.08T + 23T^{2} \)
29 \( 1 - 1.35T + 29T^{2} \)
31 \( 1 - 6.43iT - 31T^{2} \)
37 \( 1 + 7.35iT - 37T^{2} \)
41 \( 1 - 6.87iT - 41T^{2} \)
43 \( 1 - 0.209T + 43T^{2} \)
47 \( 1 + 1.35iT - 47T^{2} \)
53 \( 1 - 1.46T + 53T^{2} \)
59 \( 1 + 2.26iT - 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 - 0.438iT - 71T^{2} \)
73 \( 1 - 3.69iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 0.475iT - 83T^{2} \)
89 \( 1 + 11.0iT - 89T^{2} \)
97 \( 1 + 3.29iT - 97T^{2} \)
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

−11.10499996452431946092958016047, −10.12447742014341908199875355464, −9.202989041732763771454237106824, −8.318323015038877364978755698699, −7.29584648607673057692268462752, −6.86738215120574872649916003820, −6.02210763604710882585922543851, −4.80235020557715445685897616637, −4.00740433812267806814923646339, −2.13368966717592242971619745929, 0.63090116261909662255959524027, 2.12831735536888643355156915966, 3.20821774991765613122317316101, 4.16957085826579054765720245161, 5.42589544869629850860401160514, 6.35581724798201047799959301966, 7.970194878354092504882537998244, 8.708505092790170340840177141608, 9.523007062891960167644707593498, 10.43994665623397024993263934513

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