↑

Properties

Label 2-531-9.7-c1-0-23
Degree $2$
Conductor $531$
Sign $-0.0292 - 0.999i$
Analytic cond. $4.24005$
Root an. cond. $2.05913$
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  + (0.208 − 0.361i)2-s + (1.01 + 1.40i)3-s + (0.912 + 1.58i)4-s + (0.573 + 0.994i)5-s + (0.719 − 0.0735i)6-s + (−1.97 + 3.41i)7-s + 1.59·8-s + (−0.943 + 2.84i)9-s + 0.479·10-s + (2.89 − 5.01i)11-s + (−1.29 + 2.88i)12-s + (−1.98 − 3.43i)13-s + (0.823 + 1.42i)14-s + (−0.813 + 1.81i)15-s + (−1.49 + 2.58i)16-s − 1.31·17-s + ⋯
L(s)  = 1  + (0.147 − 0.255i)2-s + (0.585 + 0.810i)3-s + (0.456 + 0.790i)4-s + (0.256 + 0.444i)5-s + (0.293 − 0.0300i)6-s + (−0.744 + 1.29i)7-s + 0.564·8-s + (−0.314 + 0.949i)9-s + 0.151·10-s + (0.873 − 1.51i)11-s + (−0.373 + 0.832i)12-s + (−0.550 − 0.953i)13-s + (0.220 + 0.381i)14-s + (−0.210 + 0.468i)15-s + (−0.372 + 0.645i)16-s − 0.318·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0292 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0292 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.0292 - 0.999i$
Analytic conductor: \(4.24005\)
Root analytic conductor: \(2.05913\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :1/2),\ -0.0292 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41433 + 1.45628i\)
\(L(\frac12)\) \(\approx\) \(1.41433 + 1.45628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.01 - 1.40i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.208 + 0.361i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.573 - 0.994i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.97 - 3.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.89 + 5.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.98 + 3.43i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 - 0.413T + 19T^{2} \)
23 \( 1 + (-1.59 - 2.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.434 + 0.752i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.19 + 5.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.30T + 37T^{2} \)
41 \( 1 + (-1.69 - 2.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.47 + 2.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.44 - 7.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
61 \( 1 + (1.62 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.48T + 71T^{2} \)
73 \( 1 + 6.01T + 73T^{2} \)
79 \( 1 + (1.50 - 2.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.49 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.89T + 89T^{2} \)
97 \( 1 + (-3.93 + 6.81i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.17626487778160169365390478244, −10.17547152163346386333520670446, −9.220438819348571606781560621832, −8.592827270961407938092627325073, −7.67582453984193262458355551106, −6.34038202533466660539174798057, −5.56162825861269535602514184189, −4.01050653275948319383437895302, −3.00108177732715866259247558626, −2.57025349697313839276401195922, 1.14530729932232757509248943257, 2.19799915708060566674237837823, 3.91781296061666673929730918817, 4.92753498367652824649729171348, 6.49846403283708296462030833216, 6.90775991496668155267002338634, 7.46892208643611663982727624096, 9.087005635760722511157794155511, 9.592785348734265511839510064762, 10.44287970540606345092460391119

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