Properties

Label 2-570-285.263-c1-0-8
Degree $2$
Conductor $570$
Sign $0.639 - 0.769i$
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  + (−0.0871 − 0.996i)2-s + (1.40 + 1.01i)3-s + (−0.984 + 0.173i)4-s + (−2.23 + 0.130i)5-s + (0.887 − 1.48i)6-s + (−0.693 − 0.185i)7-s + (0.258 + 0.965i)8-s + (0.945 + 2.84i)9-s + (0.324 + 2.21i)10-s + (0.957 + 0.552i)11-s + (−1.55 − 0.754i)12-s + (1.59 + 3.42i)13-s + (−0.124 + 0.707i)14-s + (−3.26 − 2.07i)15-s + (0.939 − 0.342i)16-s + (0.308 + 3.53i)17-s + ⋯
L(s)  = 1  + (−0.0616 − 0.704i)2-s + (0.810 + 0.585i)3-s + (−0.492 + 0.0868i)4-s + (−0.998 + 0.0583i)5-s + (0.362 − 0.607i)6-s + (−0.262 − 0.0702i)7-s + (0.0915 + 0.341i)8-s + (0.315 + 0.948i)9-s + (0.102 + 0.699i)10-s + (0.288 + 0.166i)11-s + (−0.450 − 0.217i)12-s + (0.443 + 0.950i)13-s + (−0.0333 + 0.188i)14-s + (−0.843 − 0.536i)15-s + (0.234 − 0.0855i)16-s + (0.0749 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.639 - 0.769i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.639 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20379 + 0.564793i\)
\(L(\frac12)\) \(\approx\) \(1.20379 + 0.564793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0871 + 0.996i)T \)
3 \( 1 + (-1.40 - 1.01i)T \)
5 \( 1 + (2.23 - 0.130i)T \)
19 \( 1 + (-0.650 - 4.31i)T \)
good7 \( 1 + (0.693 + 0.185i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.957 - 0.552i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.59 - 3.42i)T + (-8.35 + 9.95i)T^{2} \)
17 \( 1 + (-0.308 - 3.53i)T + (-16.7 + 2.95i)T^{2} \)
23 \( 1 + (-0.482 - 0.337i)T + (7.86 + 21.6i)T^{2} \)
29 \( 1 + (0.211 - 0.177i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.29 + 2.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.45 + 1.45i)T - 37iT^{2} \)
41 \( 1 + (-2.75 - 7.57i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (5.18 - 3.63i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (-4.28 - 0.374i)T + (46.2 + 8.16i)T^{2} \)
53 \( 1 + (3.77 + 2.64i)T + (18.1 + 49.8i)T^{2} \)
59 \( 1 + (5.58 + 4.68i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.14 + 6.46i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-0.128 + 1.47i)T + (-65.9 - 11.6i)T^{2} \)
71 \( 1 + (-14.0 - 2.47i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (2.29 + 1.06i)T + (46.9 + 55.9i)T^{2} \)
79 \( 1 + (2.66 + 7.32i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.22 - 1.66i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.78 - 2.10i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-15.8 + 1.38i)T + (95.5 - 16.8i)T^{2} \)
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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

−10.89402279604625958939698983874, −9.942115712838183297410875032572, −9.220720493330660188336607157343, −8.321231295254973036819463368095, −7.69067704404888958512263021681, −6.38176098312653718040111907155, −4.76172821441940559291816773593, −3.91043692045237017027583882912, −3.29589934687525087102566542112, −1.76359114151795956438597614585, 0.74832542770664070951705119456, 2.89060825883479421122723778809, 3.79826548372638044186770229995, 5.05261977418438751074243048715, 6.36188825890858971928517462197, 7.23349197482872307592402442639, 7.82526813401090673413787161200, 8.738801857202906706767891253076, 9.275345623281283804739107637422, 10.54701364903582093675644138885

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