↑

Properties

Label 2-531-59.58-c2-0-36
Degree $2$
Conductor $531$
Sign $0.0457 + 0.998i$
Analytic cond. $14.4687$
Root an. cond. $3.80377$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.86i·2-s + 0.531·4-s + 9.04·5-s + 1.29·7-s − 8.43i·8-s − 16.8i·10-s + 12.8i·11-s − 23.5i·13-s − 2.42i·14-s − 13.5·16-s + 10.1·17-s − 23.3·19-s + 4.80·20-s + 23.9·22-s + 9.25i·23-s + ⋯
L(s)  = 1  − 0.931i·2-s + 0.132·4-s + 1.80·5-s + 0.185·7-s − 1.05i·8-s − 1.68i·10-s + 1.17i·11-s − 1.81i·13-s − 0.172i·14-s − 0.849·16-s + 0.595·17-s − 1.22·19-s + 0.240·20-s + 1.09·22-s + 0.402i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.0457 + 0.998i$
Analytic conductor: \(14.4687\)
Root analytic conductor: \(3.80377\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :1),\ 0.0457 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.822936766\)
\(L(\frac12)\) \(\approx\) \(2.822936766\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 + (-2.69 - 58.9i)T \)
good2 \( 1 + 1.86iT - 4T^{2} \)
5 \( 1 - 9.04T + 25T^{2} \)
7 \( 1 - 1.29T + 49T^{2} \)
11 \( 1 - 12.8iT - 121T^{2} \)
13 \( 1 + 23.5iT - 169T^{2} \)
17 \( 1 - 10.1T + 289T^{2} \)
19 \( 1 + 23.3T + 361T^{2} \)
23 \( 1 - 9.25iT - 529T^{2} \)
29 \( 1 - 25.9T + 841T^{2} \)
31 \( 1 - 28.4iT - 961T^{2} \)
37 \( 1 + 22.3iT - 1.36e3T^{2} \)
41 \( 1 - 44.4T + 1.68e3T^{2} \)
43 \( 1 - 42.8iT - 1.84e3T^{2} \)
47 \( 1 + 30.6iT - 2.20e3T^{2} \)
53 \( 1 + 41.9T + 2.80e3T^{2} \)
61 \( 1 - 31.9iT - 3.72e3T^{2} \)
67 \( 1 + 94.0iT - 4.48e3T^{2} \)
71 \( 1 + 5.78T + 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 105.T + 6.24e3T^{2} \)
83 \( 1 + 3.14iT - 6.88e3T^{2} \)
89 \( 1 + 21.8iT - 7.92e3T^{2} \)
97 \( 1 - 126. iT - 9.40e3T^{2} \)
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

−10.23993593865699513124674918871, −10.02351155920676200663668257008, −9.032402459312483492135956065560, −7.72148040296870142979689932148, −6.58779971540927388560605458455, −5.77246185873625599864335260527, −4.73020649532385070993600277620, −3.08495491331383201447464600573, −2.21933556309003073283490323151, −1.23677401433421655836910859843, 1.66404160109662437805278722918, 2.62694661996370847168638495398, 4.54381259282556123655148432054, 5.67975885328750742509890383738, 6.28231401871498061079755554925, 6.83199538141474432331761020682, 8.253537665347948529900828914537, 8.932870820496800976195496983286, 9.810897856241800411534103527491, 10.82753764025165052748615845108

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