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Properties

Label 2-570-95.18-c1-0-10
Degree $2$
Conductor $570$
Sign $0.883 + 0.468i$
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  + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + (2.23 + 0.0328i)5-s − 1.00·6-s + (−3.17 + 3.17i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−1.55 − 1.60i)10-s + 1.49·11-s + (0.707 + 0.707i)12-s + (2.38 − 2.38i)13-s + 4.49·14-s + (1.60 − 1.55i)15-s − 1.00·16-s + (0.853 − 0.853i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (0.999 + 0.0146i)5-s − 0.408·6-s + (−1.20 + 1.20i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.492 − 0.507i)10-s + 0.451·11-s + (0.204 + 0.204i)12-s + (0.662 − 0.662i)13-s + 1.20·14-s + (0.414 − 0.402i)15-s − 0.250·16-s + (0.206 − 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.883 + 0.468i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.883 + 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42225 - 0.354031i\)
\(L(\frac12)\) \(\approx\) \(1.42225 - 0.354031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-2.23 - 0.0328i)T \)
19 \( 1 + (-4.34 + 0.349i)T \)
good7 \( 1 + (3.17 - 3.17i)T - 7iT^{2} \)
11 \( 1 - 1.49T + 11T^{2} \)
13 \( 1 + (-2.38 + 2.38i)T - 13iT^{2} \)
17 \( 1 + (-0.853 + 0.853i)T - 17iT^{2} \)
23 \( 1 + (-4.67 - 4.67i)T + 23iT^{2} \)
29 \( 1 - 5.40T + 29T^{2} \)
31 \( 1 - 0.364iT - 31T^{2} \)
37 \( 1 + (4.29 + 4.29i)T + 37iT^{2} \)
41 \( 1 - 1.79iT - 41T^{2} \)
43 \( 1 + (-0.623 - 0.623i)T + 43iT^{2} \)
47 \( 1 + (5.24 - 5.24i)T - 47iT^{2} \)
53 \( 1 + (5.27 - 5.27i)T - 53iT^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (0.989 + 0.989i)T + 67iT^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 + (11.2 + 11.2i)T + 73iT^{2} \)
79 \( 1 + 7.95T + 79T^{2} \)
83 \( 1 + (-3.94 - 3.94i)T + 83iT^{2} \)
89 \( 1 + 9.30T + 89T^{2} \)
97 \( 1 + (8.79 + 8.79i)T + 97iT^{2} \)
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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

−10.46850168087741306169503529246, −9.457937165981291891359624830311, −9.263533250206875004213950235903, −8.316299968040881627315482943049, −7.05597725533798848515240860593, −6.18056657520039703771970356153, −5.33481796556140224198959655991, −3.30762162546295301578995866903, −2.74499917624693505432484298531, −1.32761588743658772649052663833, 1.20898469403615763449748684487, 2.99198246664339686062714499236, 4.13334862580131566155326392780, 5.40026974550572834853092914061, 6.70406422480048881667435098901, 6.83895258205442986276539195922, 8.370827417621932970399901895494, 9.140025622103034671226216899738, 9.972626297700905143218284409902, 10.26209901681779087216321514989

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