↑

Properties

Label 1-89-89.26-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.163 + 0.986i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (0.841 + 0.540i)2-s + (−0.936 + 0.349i)3-s + (0.415 + 0.909i)4-s + (0.989 + 0.142i)5-s + (−0.977 − 0.212i)6-s + (0.800 − 0.599i)7-s + (−0.142 + 0.989i)8-s + (0.755 − 0.654i)9-s + (0.755 + 0.654i)10-s + (0.142 + 0.989i)11-s + (−0.707 − 0.707i)12-s + (0.349 + 0.936i)13-s + (0.997 − 0.0713i)14-s + (−0.977 + 0.212i)15-s + (−0.654 + 0.755i)16-s + (−0.540 − 0.841i)17-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.936 + 0.349i)3-s + (0.415 + 0.909i)4-s + (0.989 + 0.142i)5-s + (−0.977 − 0.212i)6-s + (0.800 − 0.599i)7-s + (−0.142 + 0.989i)8-s + (0.755 − 0.654i)9-s + (0.755 + 0.654i)10-s + (0.142 + 0.989i)11-s + (−0.707 − 0.707i)12-s + (0.349 + 0.936i)13-s + (0.997 − 0.0713i)14-s + (−0.977 + 0.212i)15-s + (−0.654 + 0.755i)16-s + (−0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.163 + 0.986i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.163 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.632052923 + 1.924607927i\)
\(L(\frac12)\) \(\approx\) \(1.632052923 + 1.924607927i\)
\(L(1)\) \(\approx\) \(1.400302635 + 0.8645716883i\)
\(L(1)\) \(\approx\) \(1.400302635 + 0.8645716883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.936 + 0.349i)T \)
5 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (0.800 - 0.599i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (0.349 + 0.936i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (-0.997 - 0.0713i)T \)
23 \( 1 + (-0.0713 + 0.997i)T \)
29 \( 1 + (0.599 + 0.800i)T \)
31 \( 1 + (-0.0713 - 0.997i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.349 + 0.936i)T \)
43 \( 1 + (0.599 - 0.800i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (-0.909 - 0.415i)T \)
59 \( 1 + (0.936 + 0.349i)T \)
61 \( 1 + (-0.877 - 0.479i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.989 + 0.142i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.755 - 0.654i)T \)
83 \( 1 + (0.977 + 0.212i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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

−30.05520993158400485018993286667, −28.98890517634938304713168208879, −28.31085658975654358692523446528, −27.29189757504709097469533600900, −25.19775163671368540737049913128, −24.47044391421815548112564184330, −23.60431548159787727377820669087, −22.269804872066729268948196786609, −21.64548431744707038905473485964, −20.78529292349543943033843415312, −19.14714772405507524877819073311, −18.13067905261311024724441593555, −17.11461331657367218646355439521, −15.672257424454742334526851940197, −14.34578360415605328154404347471, −13.192419362411508385028908670517, −12.37617806634154603250238709480, −11.052553660118636463044051882146, −10.39145197327910865471220155344, −8.51813645986015911852059613554, −6.32213544813320090785008900571, −5.74650480135223723527425116219, −4.55515679102469011321868072686, −2.41945737059094754278144158183, −1.10614791907461548109146164094, 1.90466508637962537169439089303, 4.190703017321729371625219501638, 5.03934324113325102311851605307, 6.346227245472486322075808610768, 7.25363122556603709747818336571, 9.26408543754343602269747708249, 10.747220341409402763948321211301, 11.72847228762607887768527151361, 13.06484310713018133085536132008, 14.13709087445930071966676277017, 15.20081356072482850528515696108, 16.5420999438507792422860726656, 17.37232979597194212625601794914, 18.07522755126544244721314697450, 20.44893085454755116275681546525, 21.29213831863554455629512567671, 22.07442428795214943674749239007, 23.219170979616083861436900953561, 23.914192952474736878928122375060, 25.15993144952034067841107391159, 26.18697525350825561497805120725, 27.33148683879154673396570433273, 28.67268871732871141550989439460, 29.66245041479555221890070337941, 30.44153966019217743444319789441

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