Properties

Label 2-570-285.47-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.998 - 0.0471i$
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  + (−0.819 − 0.573i)2-s + (−0.0360 + 1.73i)3-s + (0.342 + 0.939i)4-s + (0.368 − 2.20i)5-s + (1.02 − 1.39i)6-s + (−0.623 + 0.166i)7-s + (0.258 − 0.965i)8-s + (−2.99 − 0.125i)9-s + (−1.56 + 1.59i)10-s + (−1.95 + 1.12i)11-s + (−1.63 + 0.558i)12-s + (−3.88 + 0.340i)13-s + (0.606 + 0.220i)14-s + (3.80 + 0.717i)15-s + (−0.766 + 0.642i)16-s + (−4.32 − 3.03i)17-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (−0.0208 + 0.999i)3-s + (0.171 + 0.469i)4-s + (0.164 − 0.986i)5-s + (0.417 − 0.570i)6-s + (−0.235 + 0.0630i)7-s + (0.0915 − 0.341i)8-s + (−0.999 − 0.0416i)9-s + (−0.495 + 0.504i)10-s + (−0.588 + 0.339i)11-s + (−0.473 + 0.161i)12-s + (−1.07 + 0.0943i)13-s + (0.161 + 0.0589i)14-s + (0.982 + 0.185i)15-s + (−0.191 + 0.160i)16-s + (−1.04 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.998 - 0.0471i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.998 - 0.0471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00144490 + 0.0612468i\)
\(L(\frac12)\) \(\approx\) \(0.00144490 + 0.0612468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 + 0.573i)T \)
3 \( 1 + (0.0360 - 1.73i)T \)
5 \( 1 + (-0.368 + 2.20i)T \)
19 \( 1 + (-0.0260 - 4.35i)T \)
good7 \( 1 + (0.623 - 0.166i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.95 - 1.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.88 - 0.340i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (4.32 + 3.03i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (-1.63 - 0.760i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (1.38 - 7.84i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.860 - 0.860i)T + 37iT^{2} \)
41 \( 1 + (3.79 + 4.52i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (8.64 - 4.02i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (6.88 + 9.82i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (6.10 + 2.84i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.503 + 2.85i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-9.92 + 3.61i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.903 - 0.632i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-0.913 + 2.51i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.670 - 7.65i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (-6.38 - 7.60i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-2.78 + 0.746i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-8.24 - 6.92i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-7.15 + 10.2i)T + (-33.1 - 91.1i)T^{2} \)
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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

−11.05032978028888446790166079140, −9.928405745377217840652247294981, −9.678306262813770178143454796369, −8.745241171300655017033694082840, −8.008275747518589920798988043615, −6.74729375497551975852578093495, −5.26036066413126972934322332015, −4.72209216112666115654721494242, −3.42438627486341817827923550605, −2.10673072168970170237728063365, 0.03815841844299119662337670408, 2.09730983242718478436404747433, 3.00318801748458050044804215915, 4.94831624648352777927575238243, 6.22054015649811320238334188513, 6.70249764306347439728531965279, 7.57050288469515684453628376199, 8.294273962890328365971599190546, 9.377280552808140185372641528723, 10.31497887448004469139554502226

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