↑

Properties

Label 1-569-569.21-r1-0-0
Degree $1$
Conductor $569$
Sign $-0.754 - 0.655i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (0.132 + 0.991i)2-s + (−0.780 + 0.624i)3-s + (−0.964 + 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.722 − 0.691i)6-s + (0.544 + 0.839i)7-s + (−0.387 − 0.921i)8-s + (0.219 − 0.975i)9-s + (−0.930 + 0.367i)10-s + (0.995 + 0.0993i)11-s + (0.589 − 0.807i)12-s + (0.873 − 0.487i)13-s + (−0.759 + 0.650i)14-s + (−0.794 − 0.607i)15-s + (0.862 − 0.506i)16-s + (0.176 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.132 + 0.991i)2-s + (−0.780 + 0.624i)3-s + (−0.964 + 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.722 − 0.691i)6-s + (0.544 + 0.839i)7-s + (−0.387 − 0.921i)8-s + (0.219 − 0.975i)9-s + (−0.930 + 0.367i)10-s + (0.995 + 0.0993i)11-s + (0.589 − 0.807i)12-s + (0.873 − 0.487i)13-s + (−0.759 + 0.650i)14-s + (−0.794 − 0.607i)15-s + (0.862 − 0.506i)16-s + (0.176 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.754 - 0.655i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ -0.754 - 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5823793137 + 1.558091700i\)
\(L(\frac12)\) \(\approx\) \(-0.5823793137 + 1.558091700i\)
\(L(1)\) \(\approx\) \(0.5093998253 + 0.8615541753i\)
\(L(1)\) \(\approx\) \(0.5093998253 + 0.8615541753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.132 + 0.991i)T \)
3 \( 1 + (-0.780 + 0.624i)T \)
5 \( 1 + (0.240 + 0.970i)T \)
7 \( 1 + (0.544 + 0.839i)T \)
11 \( 1 + (0.995 + 0.0993i)T \)
13 \( 1 + (0.873 - 0.487i)T \)
17 \( 1 + (0.176 - 0.984i)T \)
19 \( 1 + (0.496 + 0.867i)T \)
23 \( 1 + (0.315 + 0.948i)T \)
29 \( 1 + (-0.571 + 0.820i)T \)
31 \( 1 + (-0.999 - 0.0110i)T \)
37 \( 1 + (-0.208 + 0.977i)T \)
41 \( 1 + (0.448 - 0.894i)T \)
43 \( 1 + (0.991 + 0.132i)T \)
47 \( 1 + (0.641 - 0.766i)T \)
53 \( 1 + (-0.691 + 0.722i)T \)
59 \( 1 + (0.766 + 0.641i)T \)
61 \( 1 + (-0.745 + 0.666i)T \)
67 \( 1 + (0.176 + 0.984i)T \)
71 \( 1 + (0.387 - 0.921i)T \)
73 \( 1 + (0.867 - 0.496i)T \)
79 \( 1 + (-0.0442 + 0.999i)T \)
83 \( 1 + (-0.356 - 0.934i)T \)
89 \( 1 + (0.0110 + 0.999i)T \)
97 \( 1 + (-0.294 + 0.955i)T \)
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

−22.52951767394607589839919345507, −21.62189687414276540256263099219, −20.889156481379447807592278912049, −20.02747748129612454948493238651, −19.36058723637686899708034494424, −18.39695580092094075438857427985, −17.412786658341943493305395761246, −17.0941750292014112664233924519, −16.14181425647865246255595453516, −14.459664214904857366821966777417, −13.74334752651322613279564996968, −12.9577201020593125894816406109, −12.33859348408013661600578021840, −11.21658947555876176182443560746, −10.975831283513201128749106898246, −9.64349931592989523800262715798, −8.75446524517116463014225518255, −7.814761782892395988138077648930, −6.4626397026668198062840877880, −5.53338566688900524152046435445, −4.50423100629764117567266904150, −3.879509235723310019537160689137, −1.99647197023088395472403801115, −1.24320316996742406976774078360, −0.53046982620686096019243007423, 1.26500710977109344131227452329, 3.21032417690021382160907156920, 3.9997779366519053413018332936, 5.413796883898186241112364466220, 5.71654068722344502452012511120, 6.74764288696930122863847883167, 7.60248135102033846331286849626, 8.98213782940239254163763136920, 9.529599170479419093322065465467, 10.68871991819040957923067351257, 11.584309797878073150070122774550, 12.368163245935181387340583644278, 13.7428103624879915798630412364, 14.53060308411926403260841693052, 15.192986123780573274995711110952, 15.89356362993706362778878626704, 16.7881433381139743721166201200, 17.72366002778605519874355019002, 18.214654702299403812273648306017, 18.88227324435913518693953773854, 20.56728174552941767661690001715, 21.442604122831570248752940544566, 22.27767307885648926733481072788, 22.572787835521916420719948757026, 23.42050682988557044066131627478

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