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Properties

Label 2-570-95.27-c1-0-16
Degree $2$
Conductor $570$
Sign $0.864 + 0.503i$
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  + (0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (−0.399 − 2.20i)5-s + (0.499 + 0.866i)6-s + (−0.506 − 0.506i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (2.02 − 0.955i)10-s − 1.16·11-s + (−0.707 + 0.707i)12-s + (1.78 − 6.66i)13-s + (0.358 − 0.620i)14-s + (−0.955 − 2.02i)15-s + (0.500 − 0.866i)16-s + (1.73 − 0.465i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (−0.178 − 0.983i)5-s + (0.204 + 0.353i)6-s + (−0.191 − 0.191i)7-s + (−0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (0.639 − 0.302i)10-s − 0.350·11-s + (−0.204 + 0.204i)12-s + (0.495 − 1.84i)13-s + (0.0957 − 0.165i)14-s + (−0.246 − 0.522i)15-s + (0.125 − 0.216i)16-s + (0.421 − 0.112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.864 + 0.503i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.864 + 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62533 - 0.438755i\)
\(L(\frac12)\) \(\approx\) \(1.62533 - 0.438755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.399 + 2.20i)T \)
19 \( 1 + (-4.14 - 1.33i)T \)
good7 \( 1 + (0.506 + 0.506i)T + 7iT^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-1.78 + 6.66i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.73 + 0.465i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.93 - 0.517i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.32 + 4.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.506iT - 31T^{2} \)
37 \( 1 + (-3.08 + 3.08i)T - 37iT^{2} \)
41 \( 1 + (2.46 + 1.42i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.08 - 4.03i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.443 + 1.65i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.57 - 9.62i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.04 - 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.75 - 13.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.13 - 1.91i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.60 + 3.81i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.707 - 2.63i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.23 + 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.66 - 9.66i)T - 83iT^{2} \)
89 \( 1 + (-2.91 - 5.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.861 - 3.21i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.42994335610212114576025868210, −9.592778958485731401650540924879, −8.689565200139665950788512707936, −7.88371337140695459490891242659, −7.43562763056450857486460276912, −5.87891477120328478222746011312, −5.28025575822346343137145999614, −4.01879670159597974184458153927, −2.99860111431541705863576968416, −0.915495747571873920647610074311, 1.84623369023670429975834087830, 3.02724664495290739488325077274, 3.80632152006381108129134378034, 4.99040056597962759892770960830, 6.36545235110922743686609042314, 7.19801510534806890173931069590, 8.306136662016801922582653449761, 9.339507914819793555951532674392, 9.860841038984748998524527331857, 11.01071846941536489964477291491

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