Properties

Label 2-570-95.64-c1-0-18
Degree $2$
Conductor $570$
Sign $-0.404 + 0.914i$
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  + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.37 − 1.76i)5-s + (0.499 − 0.866i)6-s − 0.785i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.07 − 0.837i)10-s − 0.377·11-s − 0.999i·12-s + (−2.51 − 1.45i)13-s + (−0.392 − 0.680i)14-s + (−2.07 − 0.837i)15-s + (−0.5 − 0.866i)16-s + (2.45 − 1.41i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.615 − 0.788i)5-s + (0.204 − 0.353i)6-s − 0.296i·7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.655 − 0.264i)10-s − 0.113·11-s − 0.288i·12-s + (−0.697 − 0.402i)13-s + (−0.104 − 0.181i)14-s + (−0.535 − 0.216i)15-s + (−0.125 − 0.216i)16-s + (0.595 − 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.404 + 0.914i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10993 - 1.70459i\)
\(L(\frac12)\) \(\approx\) \(1.10993 - 1.70459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (1.37 + 1.76i)T \)
19 \( 1 + (2.82 + 3.32i)T \)
good7 \( 1 + 0.785iT - 7T^{2} \)
11 \( 1 + 0.377T + 11T^{2} \)
13 \( 1 + (2.51 + 1.45i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.45 + 1.41i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.86 - 4.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.01 + 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 - 0.0967iT - 37T^{2} \)
41 \( 1 + (1.43 + 2.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.371 - 0.214i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.4 - 6.04i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.00 - 3.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 + 8.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.27 - 3.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.31 - 4.22i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.86 - 10.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.95 - 1.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.785 + 1.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.98 + 1.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.61759043371806523783989109791, −9.500083203290322590186338224300, −8.786474947030077203458393565939, −7.62881545949417393491327492747, −7.07749438200636669819255718100, −5.56297679894708600013864996071, −4.72032299675113300795464002183, −3.69393050090537149817444791315, −2.58219799097758715240057134992, −0.936612609738759809173212533708, 2.37640281161383854380499734481, 3.37003789005943331196079045226, 4.31799581864338685586627577869, 5.39155062866898386048797728357, 6.62899170632847752275342114075, 7.33410076475983215452191050852, 8.250107839295989566212449728512, 9.096343993224343555549570002310, 10.35737635180636625059234910585, 10.91134724459664737624750643702

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