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Properties

Label 2-570-95.88-c1-0-2
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: $\chi_{570} (373, \cdot )$
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

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

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