↑

Properties

Label 2-531-531.58-c0-0-0
Degree $2$
Conductor $531$
Sign $-0.939 + 0.342i$
Analytic cond. $0.265003$
Root an. cond. $0.514784$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.766 + 1.32i)5-s + (−0.766 − 1.32i)7-s + (0.766 − 0.642i)9-s + (0.173 − 0.984i)12-s + (0.266 − 1.50i)15-s + (−0.499 − 0.866i)16-s − 17-s − 1.87·19-s + (−0.766 − 1.32i)20-s + (1.17 + 0.984i)21-s + (−0.673 − 1.16i)25-s + (−0.500 + 0.866i)27-s + 1.53·28-s + (0.939 + 1.62i)29-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.766 + 1.32i)5-s + (−0.766 − 1.32i)7-s + (0.766 − 0.642i)9-s + (0.173 − 0.984i)12-s + (0.266 − 1.50i)15-s + (−0.499 − 0.866i)16-s − 17-s − 1.87·19-s + (−0.766 − 1.32i)20-s + (1.17 + 0.984i)21-s + (−0.673 − 1.16i)25-s + (−0.500 + 0.866i)27-s + 1.53·28-s + (0.939 + 1.62i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(0.265003\)
Root analytic conductor: \(0.514784\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :0),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1453079335\)
\(L(\frac12)\) \(\approx\) \(0.1453079335\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + 1.87T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−11.27042724836967685774338362839, −10.74842241869516087431492183246, −10.15023321738453531788533939207, −8.928795304470651992852305266674, −7.73765428123232614015410007950, −6.76191708767876727658315263349, −6.59092686267696644506765189005, −4.58307279805616675721514929722, −3.96897052835213930311837944061, −3.08878799223429514347622345632, 0.19326494595696182075891224206, 2.01743264697831783889822290138, 4.33129018543075696682590998467, 4.86722791340025996962073109978, 5.99351274107542724328810811728, 6.45385592922162600228149118244, 8.148449297510207306451855417741, 8.822798742130401972682626939874, 9.585471971756550171267243117760, 10.64996528115053321336275523386

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