Properties

Label 2-585-1.1-c1-0-1
Degree $2$
Conductor $585$
Sign $1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 1.82·4-s − 5-s − 0.828·7-s + 1.58·8-s + 0.414·10-s − 0.585·11-s − 13-s + 0.343·14-s + 3·16-s + 4.82·17-s + 3.41·19-s + 1.82·20-s + 0.242·22-s + 1.41·23-s + 25-s + 0.414·26-s + 1.51·28-s − 5.65·29-s + 10.2·31-s − 4.41·32-s − 1.99·34-s + 0.828·35-s + 8.48·37-s − 1.41·38-s − 1.58·40-s + 8.82·41-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.313·7-s + 0.560·8-s + 0.130·10-s − 0.176·11-s − 0.277·13-s + 0.0917·14-s + 0.750·16-s + 1.17·17-s + 0.783·19-s + 0.408·20-s + 0.0517·22-s + 0.294·23-s + 0.200·25-s + 0.0812·26-s + 0.286·28-s − 1.05·29-s + 1.83·31-s − 0.780·32-s − 0.342·34-s + 0.140·35-s + 1.39·37-s − 0.229·38-s − 0.250·40-s + 1.37·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.8877276736\)
\(L(\frac12)\) \(\approx\) \(0.8877276736\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 + 0.585T + 11T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 3.07T + 43T^{2} \)
47 \( 1 + 0.828T + 47T^{2} \)
53 \( 1 - 14.4T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 7.89T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 - 8.82T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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

−10.49217856516854857244707613659, −9.714286165109304329331025621075, −9.094636004158736703950196942447, −7.931217069701776126282701574024, −7.52806058553861407863894042151, −6.06463888489816548881869814765, −5.06077252307295835194499274723, −4.08089846690771216958976744620, −2.97450618779523508374006928154, −0.897256041534199990641944032128, 0.897256041534199990641944032128, 2.97450618779523508374006928154, 4.08089846690771216958976744620, 5.06077252307295835194499274723, 6.06463888489816548881869814765, 7.52806058553861407863894042151, 7.931217069701776126282701574024, 9.094636004158736703950196942447, 9.714286165109304329331025621075, 10.49217856516854857244707613659

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