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Properties

Label 2-570-15.8-c1-0-0
Degree $2$
Conductor $570$
Sign $-0.818 + 0.574i$
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 + (−1.55 − 0.752i)3-s + 1.00i·4-s + (−1.50 + 1.64i)5-s + (−0.570 − 1.63i)6-s + (0.306 − 0.306i)7-s + (−0.707 + 0.707i)8-s + (1.86 + 2.34i)9-s + (−2.23 + 0.0989i)10-s + 0.944i·11-s + (0.752 − 1.55i)12-s + (−4.86 − 4.86i)13-s + 0.433·14-s + (3.59 − 1.43i)15-s − 1.00·16-s + (−2.18 − 2.18i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.900 − 0.434i)3-s + 0.500i·4-s + (−0.675 + 0.737i)5-s + (−0.232 − 0.667i)6-s + (0.115 − 0.115i)7-s + (−0.250 + 0.250i)8-s + (0.622 + 0.782i)9-s + (−0.706 + 0.0312i)10-s + 0.284i·11-s + (0.217 − 0.450i)12-s + (−1.34 − 1.34i)13-s + 0.115·14-s + (0.928 − 0.370i)15-s − 0.250·16-s + (−0.529 − 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.818 + 0.574i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.818 + 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0160793 - 0.0509120i\)
\(L(\frac12)\) \(\approx\) \(0.0160793 - 0.0509120i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 0.752i)T \)
5 \( 1 + (1.50 - 1.64i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.306 + 0.306i)T - 7iT^{2} \)
11 \( 1 - 0.944iT - 11T^{2} \)
13 \( 1 + (4.86 + 4.86i)T + 13iT^{2} \)
17 \( 1 + (2.18 + 2.18i)T + 17iT^{2} \)
23 \( 1 + (5.00 - 5.00i)T - 23iT^{2} \)
29 \( 1 - 1.77T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + (3.09 - 3.09i)T - 37iT^{2} \)
41 \( 1 + 6.08iT - 41T^{2} \)
43 \( 1 + (3.19 + 3.19i)T + 43iT^{2} \)
47 \( 1 + (-4.22 - 4.22i)T + 47iT^{2} \)
53 \( 1 + (-5.08 + 5.08i)T - 53iT^{2} \)
59 \( 1 + 7.52T + 59T^{2} \)
61 \( 1 - 4.53T + 61T^{2} \)
67 \( 1 + (-7.65 + 7.65i)T - 67iT^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + (-2.53 - 2.53i)T + 73iT^{2} \)
79 \( 1 - 9.50iT - 79T^{2} \)
83 \( 1 + (7.84 - 7.84i)T - 83iT^{2} \)
89 \( 1 - 8.88T + 89T^{2} \)
97 \( 1 + (0.722 - 0.722i)T - 97iT^{2} \)
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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

−11.36473517694940339066528653601, −10.61914317803897419199396660003, −9.710962136191188101638866016381, −8.116994812165584409684221047805, −7.35478978728073264846781086089, −6.96783630620110549571673427319, −5.70134085734100574676496003216, −4.99052910455042467921787919540, −3.80821339313921995086046415122, −2.42096114516693948624743224323, 0.02717869124905364437111253609, 1.91003531852134682843075012238, 3.80605868730701484770937047415, 4.49200373909906373493831368038, 5.25723252465387029891559134957, 6.35414824769677139874490543938, 7.35155917532900367779797271293, 8.711211919422604896356338900573, 9.480302731060660690149663421011, 10.42955115356432566003476671857

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