↑

Properties

Label 2-532-133.37-c0-0-1
Degree $2$
Conductor $532$
Sign $0.895 + 0.444i$
Analytic cond. $0.265502$
Root an. cond. $0.515269$
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  + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−1 − 1.73i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)35-s − 43-s + (0.499 + 0.866i)45-s + (−1 + 1.73i)47-s + 49-s − 1.99·55-s + (−1 + 1.73i)61-s + (−0.5 + 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−1 − 1.73i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)35-s − 43-s + (0.499 + 0.866i)45-s + (−1 + 1.73i)47-s + 49-s − 1.99·55-s + (−1 + 1.73i)61-s + (−0.5 + 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.265502\)
Root analytic conductor: \(0.515269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9773239855\)
\(L(\frac12)\) \(\approx\) \(0.9773239855\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−10.80586181852080632349541324149, −10.43417693083198184373789194958, −8.918145196570350113020173579221, −8.297768829904437201319863794593, −7.84483887038179014517738524124, −6.02470916889726814118017157017, −5.44669032102847998337813284477, −4.57714236045013349233443617036, −2.97858536469963990867659022587, −1.54675859337263129998469221624, 2.02971190226874062559952940153, 3.07235717316408020694215280717, 4.66490704319140395632118686786, 5.43626596634302760288955360460, 6.77891612716172954239535247220, 7.33882750159943203159656852194, 8.431050937786117325797763253107, 9.564291806996441187246636934588, 10.16644384465903336371900381510, 11.13959054975195862916308810271

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