Properties

Label 2-570-57.56-c1-0-21
Degree $2$
Conductor $570$
Sign $0.662 + 0.749i$
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 + 3.41·7-s + 8-s + (−1.00 − 2.82i)9-s i·10-s + 2.58i·11-s + (1 − 1.41i)12-s + 6.24i·13-s + 3.41·14-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 + 1.29·7-s + 0.353·8-s + (−0.333 − 0.942i)9-s − 0.316i·10-s + 0.779i·11-s + (0.288 − 0.408i)12-s + 1.73i·13-s + 0.912·14-s + (−0.365 − 0.258i)15-s + 0.250·16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.662 + 0.749i$
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: \((2,\ 570,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62226 - 1.18198i\)
\(L(\frac12)\) \(\approx\) \(2.62226 - 1.18198i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
13 \( 1 - 6.24iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 9.07T + 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 - 6.24iT - 37T^{2} \)
41 \( 1 - 3.07T + 41T^{2} \)
43 \( 1 - 9.41T + 43T^{2} \)
47 \( 1 - 8.82iT - 47T^{2} \)
53 \( 1 + 8.82T + 53T^{2} \)
59 \( 1 + 7.17T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 9.31iT - 79T^{2} \)
83 \( 1 + 7.17iT - 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + 9.41iT - 97T^{2} \)
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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

−11.05108002096905571874987904490, −9.499443160654880028387070724353, −8.769348124728906008607094931375, −7.80202242227694047962297532147, −7.05893811535262297355140259962, −6.13802179362875063779122290392, −4.72734722309184612335936219697, −4.19930328807414538843215952970, −2.39148028283665903808328174644, −1.62043900200395382227800313105, 2.02476914482624975678854849834, 3.29687294806212818937218886176, 4.06691870951529108305198035204, 5.33073275625658685155226248282, 5.82422646652259103790099859648, 7.60048286717238886568321299370, 8.028681042558206351001899027854, 9.028307458576242894665517722794, 10.32683681460412669327763896967, 10.88839689649656586309831289134

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