↑

Properties

Label 1-532-532.47-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.401 + 0.916i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)5-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (−0.939 − 0.342i)29-s + 31-s + (−0.173 + 0.984i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)5-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (−0.939 − 0.342i)29-s + 31-s + (−0.173 + 0.984i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.401 + 0.916i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.401 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8998488537 + 1.376248752i\)
\(L(\frac12)\) \(\approx\) \(0.8998488537 + 1.376248752i\)
\(L(1)\) \(\approx\) \(1.135408513 + 0.6421203428i\)
\(L(1)\) \(\approx\) \(1.135408513 + 0.6421203428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.50136448363923365609542184347, −22.38864598405438349093028903890, −21.18603510385217714437343975462, −20.706187133262380608278000812938, −19.75728306608086075310874941050, −19.11551368899147388847062269173, −18.38738760696048800536820846646, −17.15943386067697429806609541053, −16.51441431330840828902825627452, −15.53873776252264423037083728785, −14.49348958852804723335698220082, −13.66836145932480077909202592637, −13.08708683615567071392991180135, −11.964718251024002536373683069739, −11.47942198975428170675505711101, −9.818112804908677134891587683048, −8.89144731225034279697663574573, −8.54269745150096210711618768206, −7.31338510274448500989092692937, −6.53366934814170718649605870001, −5.23422978599640193919949933330, −4.153881898264123683889231774113, −3.12195956042945479617452601878, −1.86511194649656044692397855649, −0.81106517560275154353150962894, 1.83054553020180381575459380213, 2.89545781895919179757378273130, 3.708113802432014216439213008359, 4.68153707591875554953770343033, 5.950718478175870452269143959376, 7.15884131532766356762164354902, 7.85990334623712063619862965585, 8.930836807287438008920592888940, 9.94573260022573624778352767056, 10.51713207540524944834429424673, 11.45191515072517645308544474939, 12.70044178196858604872156224448, 13.59436756227365394565201042200, 14.72658704663813668838128460925, 15.01462098669992216684012188061, 15.747438578894610519350051158175, 17.06137079165834443223617309402, 17.76484538914439575557144826285, 18.96257560540285069871547583379, 19.52682719308777482041869142578, 20.34913289621641993675706392202, 21.20256599674044451531630825876, 22.16895233035560369865248146976, 22.62288282005625323624506675662, 23.58437922978990821166355214925

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