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Properties

Label 2-585-13.9-c1-0-17
Degree $2$
Conductor $585$
Sign $0.0128 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 1.40i)2-s + (−0.309 − 0.535i)4-s − 5-s + (0.118 + 0.204i)7-s + 2.23·8-s + (−0.809 + 1.40i)10-s + (2.11 − 3.66i)11-s + (−1 − 3.46i)13-s + 0.381·14-s + (2.42 − 4.20i)16-s + (2.73 + 4.73i)17-s + (0.118 + 0.204i)19-s + (0.309 + 0.535i)20-s + (−3.42 − 5.93i)22-s + (4.11 − 7.13i)23-s + ⋯
L(s)  = 1  + (0.572 − 0.990i)2-s + (−0.154 − 0.267i)4-s − 0.447·5-s + (0.0446 + 0.0772i)7-s + 0.790·8-s + (−0.255 + 0.443i)10-s + (0.638 − 1.10i)11-s + (−0.277 − 0.960i)13-s + 0.102·14-s + (0.606 − 1.05i)16-s + (0.663 + 1.14i)17-s + (0.0270 + 0.0469i)19-s + (0.0690 + 0.119i)20-s + (−0.730 − 1.26i)22-s + (0.858 − 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46689 - 1.44820i\)
\(L(\frac12)\) \(\approx\) \(1.46689 - 1.44820i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + (-0.809 + 1.40i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.118 - 0.204i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.11 + 3.66i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.73 - 4.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.118 - 0.204i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 + 7.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.736 + 1.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.97 - 5.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.881 - 1.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-6.35 - 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.20 - 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.35 - 9.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.881 + 1.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.73 + 4.73i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64271676168755328733340301434, −10.05719486182582429758167488158, −8.582208120235336332416112633836, −8.055674514605799706124374871978, −6.83019166118608350711647652954, −5.68954410576967202273034888275, −4.55724722911197847853616482595, −3.55235295080124761867939471800, −2.80576361194186235420948973634, −1.14033081672855307634296520767, 1.68354101681500981589729540238, 3.54928074039369604531821340960, 4.66392160416161222538286199813, 5.26346336850704819485088613389, 6.63059812629052920961928605121, 7.15063051886026830451418508342, 7.83472348785181759999981285662, 9.205639202231114281460028411887, 9.836312274302520445828326044831, 11.09359413037000983723834205366

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