Properties

Label 2-585-13.9-c1-0-0
Degree $2$
Conductor $585$
Sign $-0.999 + 0.0136i$
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  + (−0.473 + 0.819i)2-s + (0.552 + 0.956i)4-s − 5-s + (0.781 + 1.35i)7-s − 2.93·8-s + (0.473 − 0.819i)10-s + (−0.754 + 1.30i)11-s + (−3.46 + 1.00i)13-s − 1.47·14-s + (0.285 − 0.494i)16-s + (−1.63 − 2.82i)17-s + (1.55 + 2.68i)19-s + (−0.552 − 0.956i)20-s + (−0.714 − 1.23i)22-s + (−1.46 + 2.54i)23-s + ⋯
L(s)  = 1  + (−0.334 + 0.579i)2-s + (0.276 + 0.478i)4-s − 0.447·5-s + (0.295 + 0.511i)7-s − 1.03·8-s + (0.149 − 0.259i)10-s + (−0.227 + 0.394i)11-s + (−0.960 + 0.278i)13-s − 0.395·14-s + (0.0714 − 0.123i)16-s + (−0.395 − 0.685i)17-s + (0.356 + 0.616i)19-s + (−0.123 − 0.213i)20-s + (−0.152 − 0.263i)22-s + (−0.306 + 0.530i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.999 + 0.0136i$
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: \((2,\ 585,\ (\ :1/2),\ -0.999 + 0.0136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00487919 - 0.714612i\)
\(L(\frac12)\) \(\approx\) \(0.00487919 - 0.714612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.46 - 1.00i)T \)
good2 \( 1 + (0.473 - 0.819i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.781 - 1.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.754 - 1.30i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.63 + 2.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.55 - 2.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.46 - 2.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.30 - 2.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.65T + 31T^{2} \)
37 \( 1 + (0.285 - 0.494i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.84 - 4.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.632 + 1.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.13T + 47T^{2} \)
53 \( 1 - 2.91T + 53T^{2} \)
59 \( 1 + (-6.40 - 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 + 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.74 - 8.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.610 - 1.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 7.76T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + (-0.398 + 0.690i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.90 - 5.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.39914533504032866057360545520, −10.07992692568244791215167353910, −9.197018374774659661626995747696, −8.381729491761145029826440970228, −7.50133362205321470285851815253, −7.00079331348533549743022903656, −5.76505204334947898041086879130, −4.73575277624508355585083380408, −3.39612450331145986062101219649, −2.21375426964418622656186514437, 0.41813823383855153823387318029, 2.02439802509190014260273124543, 3.23577558499124725871142205990, 4.55041612934241816305549533797, 5.62498182177589126322734301153, 6.72651609181756424780771332990, 7.64334972849307288634220556423, 8.622416044080007475510203032872, 9.571085743284041791780027471078, 10.44006969736153160824163506075

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