↑

Properties

Label 2-817-817.189-c0-0-0
Degree $2$
Conductor $817$
Sign $0.990 + 0.140i$
Analytic cond. $0.407736$
Root an. cond. $0.638542$
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  + (−0.900 + 0.433i)4-s + (−0.162 + 0.414i)5-s + (0.733 − 1.26i)7-s + (0.826 − 0.563i)9-s + (−0.658 − 0.317i)11-s + (0.623 − 0.781i)16-s + (0.698 + 1.77i)17-s + (0.826 + 0.563i)19-s + (−0.0332 − 0.443i)20-s + (−0.147 − 1.97i)23-s + (0.587 + 0.545i)25-s + (−0.109 + 1.46i)28-s + (0.406 + 0.510i)35-s + (−0.5 + 0.866i)36-s + 43-s + 0.730·44-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)4-s + (−0.162 + 0.414i)5-s + (0.733 − 1.26i)7-s + (0.826 − 0.563i)9-s + (−0.658 − 0.317i)11-s + (0.623 − 0.781i)16-s + (0.698 + 1.77i)17-s + (0.826 + 0.563i)19-s + (−0.0332 − 0.443i)20-s + (−0.147 − 1.97i)23-s + (0.587 + 0.545i)25-s + (−0.109 + 1.46i)28-s + (0.406 + 0.510i)35-s + (−0.5 + 0.866i)36-s + 43-s + 0.730·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(817\)    =    \(19 \cdot 43\)
Sign: $0.990 + 0.140i$
Analytic conductor: \(0.407736\)
Root analytic conductor: \(0.638542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{817} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 817,\ (\ :0),\ 0.990 + 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8881495827\)
\(L(\frac12)\) \(\approx\) \(0.8881495827\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 - T \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
3 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (0.162 - 0.414i)T + (-0.733 - 0.680i)T^{2} \)
7 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.955 + 0.294i)T^{2} \)
17 \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \)
23 \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.826 - 0.563i)T^{2} \)
31 \( 1 + (-0.0747 + 0.997i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (-0.365 - 0.930i)T^{2} \)
71 \( 1 + (0.988 + 0.149i)T^{2} \)
73 \( 1 + (1.23 - 0.185i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
89 \( 1 + (-0.826 + 0.563i)T^{2} \)
97 \( 1 + (-0.623 - 0.781i)T^{2} \)
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

−10.42706031403626058464111353804, −9.758443124478256483052157259692, −8.519592767394576198807670628375, −7.894560612883669569887760983788, −7.23512262649901694937241998185, −6.08428886675314730964317272727, −4.77991652838848055885194747756, −4.06193955340040926627170914949, −3.27362323420796624754352821156, −1.21918459287736450803489963423, 1.45059017827854192705769499145, 2.88442277098336299284869381772, 4.49992995239871623751670859869, 5.17003043575155596950337408212, 5.57145864037030180238181803895, 7.32005721195281957595215983298, 7.914595187238334976268183877662, 8.951347459935849903510864230324, 9.500058089554055678448582322531, 10.24103360079265543091495023629

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