↑

Properties

Label 2-585-65.29-c1-0-31
Degree $2$
Conductor $585$
Sign $-0.359 + 0.933i$
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  + (2.01 − 1.16i)2-s + (1.70 − 2.94i)4-s + (0.174 − 2.22i)5-s + (−0.473 − 0.273i)7-s − 3.26i·8-s + (−2.23 − 4.69i)10-s + (−1.98 − 3.44i)11-s + (3.05 + 1.90i)13-s − 1.26·14-s + (−0.389 − 0.674i)16-s + (0.724 + 0.418i)17-s + (−2.56 + 4.43i)19-s + (−6.27 − 4.30i)20-s + (−7.99 − 4.61i)22-s + (2.12 − 1.22i)23-s + ⋯
L(s)  = 1  + (1.42 − 0.821i)2-s + (0.850 − 1.47i)4-s + (0.0780 − 0.996i)5-s + (−0.178 − 0.103i)7-s − 1.15i·8-s + (−0.708 − 1.48i)10-s + (−0.599 − 1.03i)11-s + (0.848 + 0.528i)13-s − 0.339·14-s + (−0.0972 − 0.168i)16-s + (0.175 + 0.101i)17-s + (−0.587 + 1.01i)19-s + (−1.40 − 0.963i)20-s + (−1.70 − 0.984i)22-s + (0.442 − 0.255i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.359 + 0.933i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.359 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73601 - 2.52800i\)
\(L(\frac12)\) \(\approx\) \(1.73601 - 2.52800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.174 + 2.22i)T \)
13 \( 1 + (-3.05 - 1.90i)T \)
good2 \( 1 + (-2.01 + 1.16i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.473 + 0.273i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.98 + 3.44i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.724 - 0.418i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 1.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.89 + 5.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.43T + 31T^{2} \)
37 \( 1 + (-5.13 + 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.45 - 5.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.2 - 5.90i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.222iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + (3.31 - 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.38 + 9.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.1 - 7.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.18 - 5.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.41iT - 73T^{2} \)
79 \( 1 + 8.13T + 79T^{2} \)
83 \( 1 - 1.14iT - 83T^{2} \)
89 \( 1 + (-3.83 - 6.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.8 - 7.43i)T + (48.5 + 84.0i)T^{2} \)
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

−10.83779431413027449028363630354, −9.804467703634289504766008822493, −8.712462072720478268747332393309, −7.88237467513644871097717754331, −6.05925857866536555104974931730, −5.81506796182393867930849385142, −4.52330803004318363193028349828, −3.88175371937429399601974517527, −2.66183800055206751743274507738, −1.24952949805941535829709586904, 2.51234537310771397655343137550, 3.43938651544082369107000033772, 4.52836773270302331354502715154, 5.52072481333797974212909907337, 6.35622021662354647760660303425, 7.15201776702116357376204233067, 7.77063045549666560059579936259, 9.159575019182698623596200690337, 10.33847404182090441658417047872, 11.06273719110441263786826323376

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