↑

Properties

Label 2-570-15.2-c1-0-30
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

Related objects

Downloads

Learn more

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: $\chi_{570} (77, \cdot )$
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 \( 1 + (0.722 + 0.722i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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