Properties

Label 2-570-57.56-c1-0-2
Degree $2$
Conductor $570$
Sign $0.927 - 0.374i$
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  − 2-s + (−1 − 1.41i)3-s + 4-s i·5-s + (1 + 1.41i)6-s + 0.585·7-s − 8-s + (−1.00 + 2.82i)9-s + i·10-s + 5.41i·11-s + (−1 − 1.41i)12-s + 2.24i·13-s − 0.585·14-s + (−1.41 + i)15-s + 16-s + 2.82i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.577 − 0.816i)3-s + 0.5·4-s − 0.447i·5-s + (0.408 + 0.577i)6-s + 0.221·7-s − 0.353·8-s + (−0.333 + 0.942i)9-s + 0.316i·10-s + 1.63i·11-s + (−0.288 − 0.408i)12-s + 0.621i·13-s − 0.156·14-s + (−0.365 + 0.258i)15-s + 0.250·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766397 + 0.148984i\)
\(L(\frac12)\) \(\approx\) \(0.766397 + 0.148984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1 + 1.41i)T \)
5 \( 1 + iT \)
19 \( 1 + (-4.24 - i)T \)
good7 \( 1 - 0.585T + 7T^{2} \)
11 \( 1 - 5.41iT - 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 - 6.82iT - 31T^{2} \)
37 \( 1 - 2.24iT - 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 - 3.17iT - 47T^{2} \)
53 \( 1 - 3.17T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 4.48T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 - 2.48T + 73T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + 12.8iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 6.58iT - 97T^{2} \)
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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

−10.79837892818962674749590845992, −9.910551364808147648803213531948, −9.015711564670471186975101553585, −8.002491175347816181301896050313, −7.28563600338662704195527734285, −6.51095389077325742398044562978, −5.37646944716733884808360809484, −4.33966130192215834362048942823, −2.31484195027995799611342602018, −1.30297792593095552895869422482, 0.67745181899837192652731917851, 2.88520835534090871236014342005, 3.81374848090935712154043440781, 5.47284407434638017747440805165, 5.88946852933604129609279160345, 7.21890197136782646176877375883, 8.085965246460928168234430137183, 9.231015380080341790886389765662, 9.698302799989182034805350669125, 10.86535774528953384115027826604

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