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Properties

Label 2-585-117.103-c1-0-19
Degree $2$
Conductor $585$
Sign $0.740 - 0.671i$
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  + (−1.71 + 0.988i)2-s + (−1.13 + 1.31i)3-s + (0.954 − 1.65i)4-s + (−0.866 − 0.5i)5-s + (0.642 − 3.36i)6-s + (−3.18 + 1.84i)7-s − 0.179i·8-s + (−0.436 − 2.96i)9-s + 1.97·10-s + (1.59 − 0.921i)11-s + (1.08 + 3.12i)12-s + (−3.58 − 0.388i)13-s + (3.64 − 6.30i)14-s + (1.63 − 0.569i)15-s + (2.08 + 3.61i)16-s + 0.179·17-s + ⋯
L(s)  = 1  + (−1.21 + 0.699i)2-s + (−0.653 + 0.756i)3-s + (0.477 − 0.826i)4-s + (−0.387 − 0.223i)5-s + (0.262 − 1.37i)6-s + (−1.20 + 0.695i)7-s − 0.0635i·8-s + (−0.145 − 0.989i)9-s + 0.625·10-s + (0.481 − 0.277i)11-s + (0.313 + 0.901i)12-s + (−0.994 − 0.107i)13-s + (0.972 − 1.68i)14-s + (0.422 − 0.146i)15-s + (0.521 + 0.903i)16-s + 0.0436·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316190 + 0.121972i\)
\(L(\frac12)\) \(\approx\) \(0.316190 + 0.121972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.13 - 1.31i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (3.58 + 0.388i)T \)
good2 \( 1 + (1.71 - 0.988i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.18 - 1.84i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.59 + 0.921i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.179T + 17T^{2} \)
19 \( 1 + 0.475iT - 19T^{2} \)
23 \( 1 + (-3.15 + 5.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.00188 + 0.00326i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.31 + 0.758i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.39iT - 37T^{2} \)
41 \( 1 + (-1.53 - 0.885i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.10 - 1.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.86 + 4.54i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.08T + 53T^{2} \)
59 \( 1 + (-3.80 - 2.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.04 - 5.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 - 4.31iT - 73T^{2} \)
79 \( 1 + (5.42 + 9.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.1 + 8.75i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.06iT - 89T^{2} \)
97 \( 1 + (-9.54 + 5.50i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39295346269882226121066792284, −9.744937214713654914534144801538, −9.100932603974091615277794382709, −8.457278546649196059545141593210, −7.13556261298571204762397923230, −6.46975538378085513654539890692, −5.57952253943143179827681674687, −4.30544512372909210684478605979, −3.04732296305035054747933837424, −0.49414581134291501142146685065, 0.76067142099262618999539032514, 2.22348014350513760672150747047, 3.52882518285558458926860821186, 5.10796358458104325722313619453, 6.42460869023461930667134308749, 7.29017435328097942736432825962, 7.76980818742203950784460218823, 9.167337746340582452760572475696, 9.745316518999899053833173401603, 10.64261885342177826444294016115

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