↑

Properties

Label 1-532-532.251-r0-0-0
Degree $1$
Conductor $532$
Sign $0.990 + 0.135i$
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  + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (−0.766 − 0.642i)17-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.766 − 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + 37-s − 39-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (−0.766 − 0.642i)17-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.766 − 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + 37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.990 + 0.135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9739306743 + 0.06611425642i\)
\(L(\frac12)\) \(\approx\) \(0.9739306743 + 0.06611425642i\)
\(L(1)\) \(\approx\) \(0.8338215380 + 0.1060781075i\)
\(L(1)\) \(\approx\) \(0.8338215380 + 0.1060781075i\)

Euler product

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

−23.60015131227388431579534277751, −22.73751412271711495419595125953, −21.84962180720616663433118375448, −21.002192091951490977971529112572, −19.972224245161922079527590810592, −19.40202217163614884685828445066, −17.99508689121949161191019587276, −17.67523871872705114364641443485, −16.70579401294181960155486658207, −15.966008319021659605885644062590, −15.20645918780962022939292115436, −13.7698231636163169120715556854, −12.90183434043923986245655284078, −12.35213028119006435684784383706, −11.44633073342753747347736020295, −10.59870257363554087220793613322, −9.48574467234497100974154921614, −8.52275160544482899853968399746, −7.51149889156962955208267596902, −6.49669831373872838020719527115, −5.59243807945479613796376546224, −4.66641650482774409916750738014, −3.81545523891339101156189908388, −1.889658431589340705349144923467, −1.02489938672653252475056393817, 0.77775800067618030905700644204, 2.47312273314242761554378192980, 3.73108482758546401917608718985, 4.47741420206929623932080652487, 6.00386802336902032826765300640, 6.33938664418593340220007465472, 7.38871489445945409442207594152, 8.6884329674622886807399872731, 9.70264245088472993984318945422, 10.77307477135310195868662538278, 11.241590466019613777639589994527, 11.94986085354660635172495148313, 13.24618431105365375975943964432, 14.123764185054512790547113223744, 15.09478454935774770072442736278, 15.978094223269921913941728560899, 16.57929902512672471574177676461, 17.67926801222243718241646194302, 18.39517383869897936063569623204, 18.999232335250385265459047874, 20.18588673732791493576750998420, 21.25730273494986955745523166831, 21.94490075214854422118964799395, 22.62847539834765597296120175874, 23.27938429646439163081362863638

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