↑

Properties

Label 1-319-319.137-r1-0-0
Degree $1$
Conductor $319$
Sign $0.0834 + 0.996i$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
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  + (0.998 + 0.0448i)2-s + (0.0896 + 0.995i)3-s + (0.995 + 0.0896i)4-s + (0.963 − 0.266i)5-s + (0.0448 + 0.998i)6-s + (0.858 + 0.512i)7-s + (0.990 + 0.134i)8-s + (−0.983 + 0.178i)9-s + (0.974 − 0.222i)10-s + i·12-s + (−0.473 + 0.880i)13-s + (0.834 + 0.550i)14-s + (0.351 + 0.936i)15-s + (0.983 + 0.178i)16-s + (−0.587 + 0.809i)17-s + (−0.990 + 0.134i)18-s + ⋯
L(s)  = 1  + (0.998 + 0.0448i)2-s + (0.0896 + 0.995i)3-s + (0.995 + 0.0896i)4-s + (0.963 − 0.266i)5-s + (0.0448 + 0.998i)6-s + (0.858 + 0.512i)7-s + (0.990 + 0.134i)8-s + (−0.983 + 0.178i)9-s + (0.974 − 0.222i)10-s + i·12-s + (−0.473 + 0.880i)13-s + (0.834 + 0.550i)14-s + (0.351 + 0.936i)15-s + (0.983 + 0.178i)16-s + (−0.587 + 0.809i)17-s + (−0.990 + 0.134i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0834 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0834 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.0834 + 0.996i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ 0.0834 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.731343691 + 3.431831482i\)
\(L(\frac12)\) \(\approx\) \(3.731343691 + 3.431831482i\)
\(L(1)\) \(\approx\) \(2.304852176 + 1.080338294i\)
\(L(1)\) \(\approx\) \(2.304852176 + 1.080338294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.998 + 0.0448i)T \)
3 \( 1 + (0.0896 + 0.995i)T \)
5 \( 1 + (0.963 - 0.266i)T \)
7 \( 1 + (0.858 + 0.512i)T \)
13 \( 1 + (-0.473 + 0.880i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.512 - 0.858i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.998 + 0.0448i)T \)
37 \( 1 + (-0.722 - 0.691i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (0.722 - 0.691i)T \)
53 \( 1 + (-0.0448 + 0.998i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.657 - 0.753i)T \)
67 \( 1 + (0.900 + 0.433i)T \)
71 \( 1 + (-0.983 - 0.178i)T \)
73 \( 1 + (0.351 + 0.936i)T \)
79 \( 1 + (-0.178 - 0.983i)T \)
83 \( 1 + (-0.393 + 0.919i)T \)
89 \( 1 + (-0.781 - 0.623i)T \)
97 \( 1 + (-0.657 + 0.753i)T \)
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

−24.69437491134433098930045532461, −23.93840976047666792092873858304, −22.816475523516875604910079444748, −22.38089888261628722814306811410, −20.92883620008229345302533033881, −20.60985359307344006068387780143, −19.48035927140154984825765333559, −18.383028262109391317206892154347, −17.45723592222248725778462512870, −16.80259041112760963534846062078, −15.17808080306657754195316923507, −14.364146403711484442459516753949, −13.74785029428701902507799577015, −12.96693423292272161056623907896, −12.05974437976065548768939482486, −11.000549977561499941936677203832, −10.17026536929410159738235841896, −8.47131247140990767074674409254, −7.40012649562419812690444908148, −6.598519464871838589201793657615, −5.607304170005956006915606090808, −4.66630437112825171693852705600, −2.995891083833597268500557044984, −2.15051589397466186099724055696, −1.06064486099966101268531073219, 1.80510377590950276848701994573, 2.664097758457756813102041443, 4.17241373243504755897730867529, 4.937865730811044947440818585507, 5.69266531423891901692609571257, 6.81802423045729301189474413790, 8.42707252226436734240882581746, 9.29254508067150873133950904716, 10.51033602181433630440783739165, 11.276630648967952810120017589028, 12.27851369996036352797881724369, 13.511021778362858507658947154344, 14.220886953958403164901787982410, 15.07707832467931632425340381167, 15.745710341447315148233273205830, 17.11509298531525936625794257144, 17.34383672486142068359779109172, 19.139717654429806242287837639665, 20.21789389538813703166181381714, 21.0768785761789182613463977157, 21.69964927517475484841545145370, 21.96343899957155161861482473822, 23.3410195814540744817058246267, 24.251473152386206562158401521719, 25.026721952895272844369239004833

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