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Properties

Label 2-585-65.9-c1-0-23
Degree $2$
Conductor $585$
Sign $0.997 - 0.0762i$
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  + (1.35 + 0.783i)2-s + (0.226 + 0.391i)4-s + (1.90 − 1.16i)5-s + (0.331 − 0.191i)7-s − 2.42i·8-s + (3.50 − 0.0880i)10-s + (−2.30 + 3.99i)11-s + (3.53 + 0.731i)13-s + 0.598·14-s + (2.35 − 4.07i)16-s + (4.29 − 2.47i)17-s + (−2.05 − 3.55i)19-s + (0.888 + 0.483i)20-s + (−6.26 + 3.61i)22-s + (5.72 + 3.30i)23-s + ⋯
L(s)  = 1  + (0.959 + 0.553i)2-s + (0.113 + 0.195i)4-s + (0.853 − 0.521i)5-s + (0.125 − 0.0722i)7-s − 0.856i·8-s + (1.10 − 0.0278i)10-s + (−0.695 + 1.20i)11-s + (0.979 + 0.202i)13-s + 0.159·14-s + (0.587 − 1.01i)16-s + (1.04 − 0.600i)17-s + (−0.470 − 0.815i)19-s + (0.198 + 0.108i)20-s + (−1.33 + 0.770i)22-s + (1.19 + 0.689i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.997 - 0.0762i$
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: \((2,\ 585,\ (\ :1/2),\ 0.997 - 0.0762i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68083 + 0.102300i\)
\(L(\frac12)\) \(\approx\) \(2.68083 + 0.102300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 1.16i)T \)
13 \( 1 + (-3.53 - 0.731i)T \)
good2 \( 1 + (-1.35 - 0.783i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-0.331 + 0.191i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.30 - 3.99i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.29 + 2.47i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.05 + 3.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.72 - 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.65 + 4.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.80T + 31T^{2} \)
37 \( 1 + (-1.44 - 0.835i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.22 - 10.7i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.10 - 4.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.71iT - 47T^{2} \)
53 \( 1 - 2.07iT - 53T^{2} \)
59 \( 1 + (-4.11 - 7.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.07 - 1.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.56 + 2.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 + 1.45T + 79T^{2} \)
83 \( 1 - 2.02iT - 83T^{2} \)
89 \( 1 + (-2.60 + 4.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.7 - 9.10i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.62343615338486573479042963432, −9.726776924736644236950250025475, −9.148948194874729408406271026872, −7.81346430639585864195398439488, −6.87677643025971190809984933157, −5.97815658914089959468218325431, −5.07883239059388348664669845277, −4.54924433894837507613008682458, −3.07264585034174556495657086428, −1.41278788129620308641705955135, 1.77059975870408998477080514983, 3.11501817344077208664224946427, 3.67518979758032858069209687995, 5.40767980274553551986318272620, 5.61084694784230759955542849616, 6.85386686362155197717121515980, 8.287921961471144266207260367570, 8.757228268158076840349715590700, 10.32500124262729005756094004110, 10.72794469429976832838835157632

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