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Properties

Label 1-451-451.379-r0-0-0
Degree $1$
Conductor $451$
Sign $0.599 + 0.800i$
Analytic cond. $2.09443$
Root an. cond. $2.09443$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + 5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + 5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(2.09443\)
Root analytic conductor: \(2.09443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 451,\ (0:\ ),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9013187217 + 0.4513245230i\)
\(L(\frac12)\) \(\approx\) \(0.9013187217 + 0.4513245230i\)
\(L(1)\) \(\approx\) \(0.8548965375 + 0.1354211967i\)
\(L(1)\) \(\approx\) \(0.8548965375 + 0.1354211967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + T \)
37 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.54015143706118413419279483369, −23.09123154844993259334951926295, −22.43561041716422873911325295620, −21.1085260106775123037382787018, −20.15338457146243974610268448054, −19.259542261311734902165844388715, −18.58903434183195860272342307293, −17.97755972543501096877764162533, −17.06415939640690085020933459975, −16.31751857157478204875596824131, −15.1736754134090316101808933559, −14.15992862490942809287684021002, −13.65981367711329965037391196696, −12.44337349905466933086440295916, −11.60509876313547383359896528370, −10.09563299552176573018366056707, −9.41874437245836890432352091431, −8.76300447882632780756645338183, −7.524901056789010042906403342201, −6.73472102755883400127291199792, −6.01860782449766436877480466320, −5.08351927056627180064124271645, −2.783618191242641580496401397575, −2.174344621264738941821607630830, −0.77268956039137997832425200836, 1.30738949948797316120109483237, 2.79998382139413273256372760433, 3.33878401436031365630762466257, 4.686266152022802832785631580993, 5.91961458963919554287636375004, 7.16880991857856002036245579300, 8.25695249499341860964112820308, 9.33088276075850107546517161275, 9.99587853527183236623120240207, 10.31877149258632009030479833382, 11.48005276351427912579572627361, 12.75437924185997453445079862835, 13.48599509351598890165510990296, 14.54611212244260214586383239050, 15.69705697501246523970258196264, 16.56153536892154372418263800920, 17.20658637156299623123761818809, 17.92249290463565576782858209057, 19.29124272229248128948367173613, 19.81711948032281486775756231249, 20.65859605201489575797553821782, 21.426594809207378752062043686192, 22.12612861118850237745885753088, 22.784405463010336037629556756831, 24.435779067690807654886445537588

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