↑

Properties

Label 1-101-101.49-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.846 - 0.532i$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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  + (0.425 + 0.904i)2-s + (−0.876 + 0.481i)3-s + (−0.637 + 0.770i)4-s + (−0.425 + 0.904i)5-s + (−0.809 − 0.587i)6-s + (−0.728 − 0.684i)7-s + (−0.968 − 0.248i)8-s + (0.535 − 0.844i)9-s − 10-s + (−0.535 + 0.844i)11-s + (0.187 − 0.982i)12-s + (0.728 − 0.684i)13-s + (0.309 − 0.951i)14-s + (−0.0627 − 0.998i)15-s + (−0.187 − 0.982i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.425 + 0.904i)2-s + (−0.876 + 0.481i)3-s + (−0.637 + 0.770i)4-s + (−0.425 + 0.904i)5-s + (−0.809 − 0.587i)6-s + (−0.728 − 0.684i)7-s + (−0.968 − 0.248i)8-s + (0.535 − 0.844i)9-s − 10-s + (−0.535 + 0.844i)11-s + (0.187 − 0.982i)12-s + (0.728 − 0.684i)13-s + (0.309 − 0.951i)14-s + (−0.0627 − 0.998i)15-s + (−0.187 − 0.982i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.846 - 0.532i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.846 - 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1312057031 + 0.4554013255i\)
\(L(\frac12)\) \(\approx\) \(-0.1312057031 + 0.4554013255i\)
\(L(1)\) \(\approx\) \(0.4009194125 + 0.5277028720i\)
\(L(1)\) \(\approx\) \(0.4009194125 + 0.5277028720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.425 + 0.904i)T \)
3 \( 1 + (-0.876 + 0.481i)T \)
5 \( 1 + (-0.425 + 0.904i)T \)
7 \( 1 + (-0.728 - 0.684i)T \)
11 \( 1 + (-0.535 + 0.844i)T \)
13 \( 1 + (0.728 - 0.684i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.187 + 0.982i)T \)
23 \( 1 + (-0.992 + 0.125i)T \)
29 \( 1 + (-0.728 + 0.684i)T \)
31 \( 1 + (0.728 + 0.684i)T \)
37 \( 1 + (0.876 + 0.481i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (-0.929 - 0.368i)T \)
47 \( 1 + (-0.929 + 0.368i)T \)
53 \( 1 + (0.637 + 0.770i)T \)
59 \( 1 + (0.187 + 0.982i)T \)
61 \( 1 + (0.637 - 0.770i)T \)
67 \( 1 + (-0.876 - 0.481i)T \)
71 \( 1 + (0.876 - 0.481i)T \)
73 \( 1 + (0.992 - 0.125i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (0.992 + 0.125i)T \)
89 \( 1 + (0.187 - 0.982i)T \)
97 \( 1 + (-0.637 + 0.770i)T \)
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

−29.08004785240410297452297324612, −28.45387097759370412043519433017, −27.902410330279055825590440506, −26.44042153526217057878368105051, −24.57687254557520937213272129280, −23.96547307513806749128982996308, −22.97616614648278824321068362516, −21.987373724448574123154250690673, −21.07537446759016043838479904054, −19.70374147949925395000719611739, −18.88618313099245613351654335921, −17.95030666568706067356462426964, −16.38437579955958096837564479871, −15.59956257352157921117384710361, −13.46687284171237519956939140736, −13.021722104894564165201646125550, −11.76341903849795253188905237813, −11.216975094411306667711920597883, −9.57922050283066746498076990889, −8.4147534278066752464827412426, −6.36463410023602254534614385779, −5.363044954490883545816623973870, −4.116391505921725849769505162198, −2.27899896624972311424750930650, −0.45534352843800957076242650264, 3.423164356323374530343326033875, 4.36149267518380856617930270094, 5.96003277427367116368428674928, 6.766457308593303265604041705160, 7.94720076894218575469675905449, 9.85980253079645543323327803218, 10.76993780622756332272519835132, 12.26170210475468203614729484952, 13.29570013783847507508496722990, 14.799845842384592755773433368558, 15.62727086786117712435157845803, 16.44085843841827722541660774264, 17.68510401009409781288518826259, 18.40629278800499082989058449699, 20.17431364263148192577409093326, 21.57233101994628274043096195321, 22.67785989905307531373871136847, 23.019162603587530648693214939823, 23.88476886830054290357917245652, 25.58856155780650474666093911880, 26.30501863634333099257119054811, 27.16147518034513336733258390468, 28.2659185422124870115887127746, 29.674515872897825710216550574618, 30.52892770977994356726367521969

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