↑

Properties

Label 1-517-517.175-r1-0-0
Degree $1$
Conductor $517$
Sign $0.925 + 0.377i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (0.334 − 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (−0.0682 − 0.997i)5-s + (−0.854 − 0.519i)6-s + (0.576 + 0.816i)7-s + (−0.854 + 0.519i)8-s + (−0.917 − 0.398i)9-s + (−0.962 − 0.269i)10-s + (−0.775 + 0.631i)12-s + (−0.682 − 0.730i)13-s + (0.962 − 0.269i)14-s + (−0.990 − 0.136i)15-s + (0.203 + 0.979i)16-s + (−0.460 + 0.887i)17-s + (−0.682 + 0.730i)18-s + ⋯
L(s)  = 1  + (0.334 − 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (−0.0682 − 0.997i)5-s + (−0.854 − 0.519i)6-s + (0.576 + 0.816i)7-s + (−0.854 + 0.519i)8-s + (−0.917 − 0.398i)9-s + (−0.962 − 0.269i)10-s + (−0.775 + 0.631i)12-s + (−0.682 − 0.730i)13-s + (0.962 − 0.269i)14-s + (−0.990 − 0.136i)15-s + (0.203 + 0.979i)16-s + (−0.460 + 0.887i)17-s + (−0.682 + 0.730i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04894678896 + 0.009606118952i\)
\(L(\frac12)\) \(\approx\) \(-0.04894678896 + 0.009606118952i\)
\(L(1)\) \(\approx\) \(0.5530437222 - 0.7800313594i\)
\(L(1)\) \(\approx\) \(0.5530437222 - 0.7800313594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.334 - 0.942i)T \)
3 \( 1 + (0.203 - 0.979i)T \)
5 \( 1 + (-0.0682 - 0.997i)T \)
7 \( 1 + (0.576 + 0.816i)T \)
13 \( 1 + (-0.682 - 0.730i)T \)
17 \( 1 + (-0.460 + 0.887i)T \)
19 \( 1 + (0.0682 - 0.997i)T \)
23 \( 1 + (-0.334 - 0.942i)T \)
29 \( 1 + (-0.682 + 0.730i)T \)
31 \( 1 + (0.203 + 0.979i)T \)
37 \( 1 + (0.962 + 0.269i)T \)
41 \( 1 + (-0.854 - 0.519i)T \)
43 \( 1 + (0.775 + 0.631i)T \)
53 \( 1 + (0.854 + 0.519i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (-0.962 + 0.269i)T \)
67 \( 1 + (-0.576 + 0.816i)T \)
71 \( 1 + (-0.334 - 0.942i)T \)
73 \( 1 + (0.917 - 0.398i)T \)
79 \( 1 + (0.990 + 0.136i)T \)
83 \( 1 + (-0.460 - 0.887i)T \)
89 \( 1 + (-0.0682 - 0.997i)T \)
97 \( 1 + (0.203 - 0.979i)T \)
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

−24.08003686699859403102979459435, −23.130395422338500428548608361695, −22.57014675777597914669390500749, −21.77424493500164197975426849892, −21.05712289430696993821279150364, −20.09171904176795976537869415818, −18.94165320056359810428197076272, −17.98051171815135271723344635497, −17.06367746258317610074708855059, −16.4779651200114191953441523592, −15.40774130410654398732563776443, −14.82976986053813743903259304752, −14.02314830031562002404813789184, −13.60698115333913523601799521465, −11.8487352528029033684006545623, −11.14869408251618122350595233681, −9.944442688922481924028859888824, −9.38451302100451802368793868751, −7.95869271890417955702583613363, −7.44284064116668324300388291578, −6.32921883372069729014397102470, −5.25934648643631897876895090712, −4.23590214583149766962473019938, −3.660420853758464259548771398238, −2.37123924377834935942017824207, 0.01121139068542516376055406305, 1.16502532022223970916781811810, 2.08785779431802149222330763688, 2.96608285686399376530854803717, 4.46793986100939767713523921157, 5.29523995548025183178409094518, 6.17694204869793869342646368044, 7.72659763015542132924948717199, 8.676462584673446278001026623980, 9.09836033616177993420562442915, 10.53801274515730113371470971312, 11.56996782128864146631483808316, 12.36575677162669426667299624207, 12.76923797651437328621677781693, 13.629518200268060532038211369651, 14.67963915684875144669543080277, 15.35758468967556178408239462125, 16.94905844904805186758841790820, 17.79850554730484557948622494860, 18.3467124247599087615431306495, 19.51496877046849411007693969361, 19.91132735130678439578080546920, 20.72240960842509686582717968282, 21.666982891689103471070916298091, 22.41157342241539119028399463931

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