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Properties

Label 2-585-65.9-c1-0-7
Degree $2$
Conductor $585$
Sign $-0.899 - 0.436i$
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  + (1.57 + 0.907i)2-s + (0.645 + 1.11i)4-s + (−1.19 + 1.88i)5-s + (−4.06 + 2.34i)7-s − 1.28i·8-s + (−3.59 + 1.88i)10-s + (−0.270 + 0.468i)11-s + (1.34 + 3.34i)13-s − 8.50·14-s + (2.45 − 4.25i)16-s + (−5.92 + 3.42i)17-s + (2.78 + 4.81i)19-s + (−2.88 − 0.118i)20-s + (−0.849 + 0.490i)22-s + (0.0291 + 0.0168i)23-s + ⋯
L(s)  = 1  + (1.11 + 0.641i)2-s + (0.322 + 0.559i)4-s + (−0.535 + 0.844i)5-s + (−1.53 + 0.886i)7-s − 0.454i·8-s + (−1.13 + 0.595i)10-s + (−0.0815 + 0.141i)11-s + (0.372 + 0.928i)13-s − 2.27·14-s + (0.614 − 1.06i)16-s + (−1.43 + 0.829i)17-s + (0.638 + 1.10i)19-s + (−0.645 − 0.0266i)20-s + (−0.181 + 0.104i)22-s + (0.00607 + 0.00350i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.899 - 0.436i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.899 - 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341389 + 1.48639i\)
\(L(\frac12)\) \(\approx\) \(0.341389 + 1.48639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.88i)T \)
13 \( 1 + (-1.34 - 3.34i)T \)
good2 \( 1 + (-1.57 - 0.907i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (4.06 - 2.34i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.270 - 0.468i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (5.92 - 3.42i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.78 - 4.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0291 - 0.0168i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.40 + 5.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.352T + 31T^{2} \)
37 \( 1 + (-6.98 - 4.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.59 - 6.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.08 + 3.51i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.99iT - 47T^{2} \)
53 \( 1 - 8.50iT - 53T^{2} \)
59 \( 1 + (6.64 + 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.60 - 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.70 - 5.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.33 - 7.50i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.05iT - 73T^{2} \)
79 \( 1 + 2.29T + 79T^{2} \)
83 \( 1 + 14.8iT - 83T^{2} \)
89 \( 1 + (3.98 - 6.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.265 - 0.153i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.32575281727323846667901498459, −10.09450442138452206700424048433, −9.447950299182342935831170171986, −8.252026312746186172249981808959, −7.03652842091139247124190381490, −6.30378723521568401946822419748, −5.97938561610725577306439334918, −4.39557884946015059201402369672, −3.64024604668323953484883528510, −2.59684998908568960486442848776, 0.58482921126923257763029725914, 2.77690346912850202162715452866, 3.58914029910500253754404093266, 4.48669409670440908022840107047, 5.35922644303120623868125489164, 6.56360581415772901727754992832, 7.52195308852851492232266085317, 8.732529439073208530301026038576, 9.486373200481512827793121768192, 10.69793202998629750572921762540

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