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Properties

Label 2-570-95.27-c1-0-19
Degree $2$
Conductor $570$
Sign $-0.876 - 0.481i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (−0.258 − 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (−0.734 − 2.11i)5-s + (−0.499 − 0.866i)6-s + (−2.57 − 2.57i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−1.84 + 1.25i)10-s − 5.12·11-s + (−0.707 + 0.707i)12-s + (−1.19 + 4.44i)13-s + (−1.81 + 3.14i)14-s + (−1.25 − 1.84i)15-s + (0.500 − 0.866i)16-s + (−2.51 + 0.673i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (−0.328 − 0.944i)5-s + (−0.204 − 0.353i)6-s + (−0.971 − 0.971i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.584 + 0.397i)10-s − 1.54·11-s + (−0.204 + 0.204i)12-s + (−0.330 + 1.23i)13-s + (−0.485 + 0.841i)14-s + (−0.324 − 0.477i)15-s + (0.125 − 0.216i)16-s + (−0.609 + 0.163i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140009 + 0.546237i\)
\(L(\frac12)\) \(\approx\) \(0.140009 + 0.546237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.734 + 2.11i)T \)
19 \( 1 + (-3.94 - 1.86i)T \)
good7 \( 1 + (2.57 + 2.57i)T + 7iT^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 + (1.19 - 4.44i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.51 - 0.673i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-5.42 - 1.45i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.79 + 8.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.03iT - 31T^{2} \)
37 \( 1 + (4.87 - 4.87i)T - 37iT^{2} \)
41 \( 1 + (5.16 + 2.98i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0709 - 0.264i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.74 + 6.50i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.54 + 9.48i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.25 + 5.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.80 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.65 + 1.78i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.61 - 3.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.73 - 6.48i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.09 - 7.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.64 - 1.64i)T - 83iT^{2} \)
89 \( 1 + (7.97 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.270 + 1.01i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.926999915571263525428284045352, −9.601833144574659958407625379989, −8.554416073249645452243561773621, −7.70716587700454884891282669777, −6.91777182161332236236267014537, −5.30825662846387346059891512025, −4.24703285627752170700687600173, −3.37567445063800607075880980901, −2.00255021950548036237469066863, −0.29962360385090300718943088696, 2.78790689319762640474143460558, 3.15313407481646950889641572237, 5.01445562261944097563545859031, 5.76353975044621239600424760614, 7.05007436064134043611785461089, 7.51230736803617686768776687556, 8.621691583713375928187749011988, 9.311273013291280048309141358562, 10.38332183044784033509241078017, 10.80435129740489701655184449645

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