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Properties

Label 2-585-117.103-c1-0-29
Degree $2$
Conductor $585$
Sign $0.841 + 0.539i$
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  + (1.62 − 0.939i)2-s + (−1.72 − 0.161i)3-s + (0.764 − 1.32i)4-s + (0.866 + 0.5i)5-s + (−2.95 + 1.35i)6-s + (−1.80 + 1.04i)7-s + 0.885i·8-s + (2.94 + 0.555i)9-s + 1.87·10-s + (4.07 − 2.35i)11-s + (−1.53 + 2.15i)12-s + (3.18 − 1.69i)13-s + (−1.95 + 3.39i)14-s + (−1.41 − 1.00i)15-s + (2.36 + 4.08i)16-s + 4.67·17-s + ⋯
L(s)  = 1  + (1.15 − 0.664i)2-s + (−0.995 − 0.0929i)3-s + (0.382 − 0.661i)4-s + (0.387 + 0.223i)5-s + (−1.20 + 0.554i)6-s + (−0.682 + 0.394i)7-s + 0.313i·8-s + (0.982 + 0.185i)9-s + 0.594·10-s + (1.22 − 0.708i)11-s + (−0.442 + 0.623i)12-s + (0.882 − 0.469i)13-s + (−0.523 + 0.906i)14-s + (−0.364 − 0.258i)15-s + (0.590 + 1.02i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.841 + 0.539i$
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: \((2,\ 585,\ (\ :1/2),\ 0.841 + 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07036 - 0.606595i\)
\(L(\frac12)\) \(\approx\) \(2.07036 - 0.606595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.161i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.18 + 1.69i)T \)
good2 \( 1 + (-1.62 + 0.939i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (1.80 - 1.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.07 + 2.35i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + 4.04iT - 19T^{2} \)
23 \( 1 + (2.18 - 3.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.28 + 2.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.329 - 0.190i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.79iT - 37T^{2} \)
41 \( 1 + (-10.4 - 6.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.00 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.347 - 0.200i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (6.53 + 3.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 + 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.00 - 5.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.42iT - 71T^{2} \)
73 \( 1 + 1.66iT - 73T^{2} \)
79 \( 1 + (0.531 + 0.920i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.57 - 0.909i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.99iT - 89T^{2} \)
97 \( 1 + (-14.4 + 8.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.14578497509521247232611118986, −10.02022555588465274718904767518, −9.164177657616050872489023100145, −7.82240792160755812230833988616, −6.27610774561175577298229999718, −6.08109935398485530757263016366, −5.07714615874668064810823796515, −3.86555495187787210373738996638, −3.04144175748577385032436421274, −1.36398354037409951189953278097, 1.28899553478155342317544889201, 3.70509080110846687140443317652, 4.24387199115980386288766404381, 5.38986136726548054097802000766, 6.23295704098480060355999609738, 6.61759928300477497289043278626, 7.65179443933908893407843467466, 9.325358449453347646365642160297, 9.896084112179248239849671799495, 10.84301495353745520046483586891

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