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Properties

Label 2-585-13.3-c1-0-0
Degree $2$
Conductor $585$
Sign $0.833 - 0.551i$
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  + (−1.31 − 2.27i)2-s + (−2.46 + 4.26i)4-s + 5-s + (−0.544 + 0.943i)7-s + 7.70·8-s + (−1.31 − 2.27i)10-s + (−2.36 − 4.08i)11-s + (−3.42 + 1.12i)13-s + 2.86·14-s + (−5.21 − 9.03i)16-s + (−2.61 + 4.52i)17-s + (−1.46 + 2.53i)19-s + (−2.46 + 4.26i)20-s + (−6.21 + 10.7i)22-s + (3.85 + 6.67i)23-s + ⋯
L(s)  = 1  + (−0.930 − 1.61i)2-s + (−1.23 + 2.13i)4-s + 0.447·5-s + (−0.205 + 0.356i)7-s + 2.72·8-s + (−0.416 − 0.720i)10-s + (−0.711 − 1.23i)11-s + (−0.950 + 0.311i)13-s + 0.766·14-s + (−1.30 − 2.25i)16-s + (−0.633 + 1.09i)17-s + (−0.335 + 0.581i)19-s + (−0.550 + 0.954i)20-s + (−1.32 + 2.29i)22-s + (0.803 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.833 - 0.551i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.833 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358624 + 0.107938i\)
\(L(\frac12)\) \(\approx\) \(0.358624 + 0.107938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.42 - 1.12i)T \)
good2 \( 1 + (1.31 + 2.27i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.544 - 0.943i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.36 + 4.08i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.61 - 4.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.46 - 2.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.85 - 6.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.655 + 1.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 + (-5.21 - 9.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.49 + 4.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.98 - 5.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.51T + 47T^{2} \)
53 \( 1 + 9.67T + 53T^{2} \)
59 \( 1 + (-1.58 + 2.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.787 - 1.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.00 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 3.04T + 79T^{2} \)
83 \( 1 + 1.40T + 83T^{2} \)
89 \( 1 + (-4.33 - 7.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.51 + 9.55i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.81515623705034574184161066639, −9.929846378544578035369820485876, −9.314886129832967110508819359272, −8.452809591820854369946408744632, −7.75887043217653222474328912137, −6.25490971084405332260812199058, −4.96185887826238128754937599667, −3.58130681870773134895965362900, −2.69122156272706521870834381911, −1.56820864620310750549355133215, 0.28254792803385107052578559093, 2.34299227922456525278932285277, 4.73889080130011844482743300849, 5.11510849481044050105272530638, 6.50335712494694467603889535597, 7.08157088922738986919996187798, 7.73707885693127703915519492892, 8.839590897395229970170934503072, 9.571763207447768577756176118244, 10.14825894520142960832186133438

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