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Properties

Label 2-585-65.29-c1-0-29
Degree $2$
Conductor $585$
Sign $-0.991 + 0.132i$
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  + (0.729 − 0.421i)2-s + (−0.644 + 1.11i)4-s + (−2.23 + 0.0545i)5-s + (0.347 + 0.200i)7-s + 2.77i·8-s + (−1.60 + 0.981i)10-s + (−2.45 − 4.24i)11-s + (−3.55 − 0.572i)13-s + 0.338·14-s + (−0.121 − 0.211i)16-s + (−6.13 − 3.54i)17-s + (1.12 − 1.95i)19-s + (1.38 − 2.53i)20-s + (−3.58 − 2.06i)22-s + (0.861 − 0.497i)23-s + ⋯
L(s)  = 1  + (0.516 − 0.297i)2-s + (−0.322 + 0.558i)4-s + (−0.999 + 0.0244i)5-s + (0.131 + 0.0758i)7-s + 0.980i·8-s + (−0.508 + 0.310i)10-s + (−0.739 − 1.28i)11-s + (−0.987 − 0.158i)13-s + 0.0903·14-s + (−0.0304 − 0.0528i)16-s + (−1.48 − 0.858i)17-s + (0.258 − 0.447i)19-s + (0.308 − 0.566i)20-s + (−0.763 − 0.440i)22-s + (0.179 − 0.103i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.991 + 0.132i$
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: \((2,\ 585,\ (\ :1/2),\ -0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00639267 - 0.0959649i\)
\(L(\frac12)\) \(\approx\) \(0.00639267 - 0.0959649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0545i)T \)
13 \( 1 + (3.55 + 0.572i)T \)
good2 \( 1 + (-0.729 + 0.421i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.347 - 0.200i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.45 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (6.13 + 3.54i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 1.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.861 + 0.497i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.94 - 6.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.30T + 31T^{2} \)
37 \( 1 + (7.62 - 4.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.65 - 4.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.74 + 1.00i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.62iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + (-1.52 + 2.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.55 + 6.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.32 - 3.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.16 + 5.48i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.01iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + (0.262 + 0.454i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.15 + 4.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71765822516852123944156263252, −9.136876121949297825120514686956, −8.519886602762249484690525379467, −7.68336906742359781648832281385, −6.83203984686424715471512783936, −5.17971213524945908222172771609, −4.67987366993465323253432669197, −3.38732735902312507997506689120, −2.68358291633164495659610882680, −0.04340211075068260690891191868, 2.11709911236928765373239647087, 3.89798630192592924410645858944, 4.58535978375816450136384424328, 5.38405080684623106378718594256, 6.74729729997783123773575362809, 7.35477310768185555842826233753, 8.388214412348035541939433686383, 9.452368991226077975741763310734, 10.27761846406627519745466888207, 11.04109672507592146035501416891

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