Properties

Label 2-570-285.62-c1-0-15
Degree $2$
Conductor $570$
Sign $-0.802 - 0.596i$
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  + (−0.422 + 0.906i)2-s + (1.13 + 1.30i)3-s + (−0.642 − 0.766i)4-s + (2.03 + 0.917i)5-s + (−1.66 + 0.478i)6-s + (−1.13 + 4.23i)7-s + (0.965 − 0.258i)8-s + (−0.412 + 2.97i)9-s + (−1.69 + 1.46i)10-s + (1.81 + 1.04i)11-s + (0.269 − 1.71i)12-s + (−3.84 − 2.69i)13-s + (−3.35 − 2.81i)14-s + (1.12 + 3.70i)15-s + (−0.173 + 0.984i)16-s + (1.56 − 3.36i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.640i)2-s + (0.656 + 0.754i)3-s + (−0.321 − 0.383i)4-s + (0.912 + 0.410i)5-s + (−0.679 + 0.195i)6-s + (−0.428 + 1.59i)7-s + (0.341 − 0.0915i)8-s + (−0.137 + 0.990i)9-s + (−0.535 + 0.461i)10-s + (0.548 + 0.316i)11-s + (0.0778 − 0.493i)12-s + (−1.06 − 0.746i)13-s + (−0.896 − 0.752i)14-s + (0.289 + 0.957i)15-s + (−0.0434 + 0.246i)16-s + (0.380 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503979 + 1.52214i\)
\(L(\frac12)\) \(\approx\) \(0.503979 + 1.52214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.422 - 0.906i)T \)
3 \( 1 + (-1.13 - 1.30i)T \)
5 \( 1 + (-2.03 - 0.917i)T \)
19 \( 1 + (1.31 + 4.15i)T \)
good7 \( 1 + (1.13 - 4.23i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.81 - 1.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.84 + 2.69i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.56 + 3.36i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (-6.58 - 0.576i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (7.59 + 2.76i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.67 - 2.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.16 - 7.16i)T + 37iT^{2} \)
41 \( 1 + (7.18 + 1.26i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.953 - 0.0834i)T + (42.3 - 7.46i)T^{2} \)
47 \( 1 + (-11.1 + 5.17i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-1.37 - 0.120i)T + (52.1 + 9.20i)T^{2} \)
59 \( 1 + (4.99 - 1.81i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.57 - 1.32i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (4.17 + 8.95i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (-6.19 + 7.38i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-5.21 - 7.45i)T + (-24.9 + 68.5i)T^{2} \)
79 \( 1 + (-5.64 - 0.995i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.61 - 6.03i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.06 - 6.04i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-6.95 - 3.24i)T + (62.3 + 74.3i)T^{2} \)
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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

−10.75999780803409593716351072644, −9.732811727755854524102206324320, −9.364105062225442360369187401466, −8.775504160132043887767425441325, −7.55045137982306982693052300799, −6.58422551773481088648017321875, −5.43522183916706369987159253353, −4.95783996945719319325027978843, −3.06968537026797699427622817206, −2.31700418973311755676483106545, 0.981264686730673896922864323742, 2.02207519785104897827941963518, 3.43643939937054504893552852601, 4.35669007631640596563607133499, 5.99370846937657423422687636668, 7.00923767559460142912723891390, 7.68819237088633442087883210360, 8.873353597646945724945710774029, 9.511122030866582494144910670508, 10.22242601908462193324778528029

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