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Properties

Label 1-55-55.19-r1-0-0
Degree $1$
Conductor $55$
Sign $-0.927 - 0.374i$
Analytic cond. $5.91057$
Root an. cond. $5.91057$
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  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s − 21-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.927 - 0.374i$
Analytic conductor: \(5.91057\)
Root analytic conductor: \(5.91057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (1:\ ),\ -0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3540187290 - 1.822628949i\)
\(L(\frac12)\) \(\approx\) \(0.3540187290 - 1.822628949i\)
\(L(1)\) \(\approx\) \(0.8808640629 - 1.050369257i\)
\(L(1)\) \(\approx\) \(0.8808640629 - 1.050369257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + T \)
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

−33.25760681906681677589542531384, −32.14819648935350084255225348961, −31.54483680282219290145593944377, −30.497596546825503317034883997089, −28.661308448280004867558630996103, −27.271927142948013006881524589129, −26.31423431137212979530861371513, −25.42520990697647202386539163472, −24.57064672768317313779925188347, −23.00554842867205504530671482120, −21.97012565669338332774483475022, −20.96379687164724453074596106673, −19.352727071598427178657867031022, −18.19916280459600225244981877417, −16.261927147919999140634575439565, −15.91652536048484263819907615084, −14.44316271628528007684629336421, −13.61765252825936355930955313793, −12.10556107994888987025349587581, −9.79164024158386072205471979467, −8.92511524497234334066237568105, −7.54011812570536257953616761716, −5.96237436497476146346471876340, −4.36836192951984431504204381236, −2.975098990054572577129445840307, 0.92455007013946867057276911201, 2.77518300968344163837178088632, 3.91982046974593582963357118391, 6.09330757112003864292508433020, 7.8957825791139815846798777036, 9.37416437398283767866015131889, 10.508623398429105164679320487048, 12.267784662041254937138433632157, 13.191021723547725927331614744262, 14.1031471916711089459343626883, 15.510385832123882922086272436204, 17.578244041942191109333986014781, 18.763642807644557627377864890139, 19.832440767424916700454731238464, 20.426189692068933160701715741779, 21.91609191773104118332147482943, 23.14759153508988826703340905942, 24.158618205392040829672967255251, 25.6926019945064988829983080968, 26.65810155627493217160572413671, 28.09140922434119222037336578745, 29.3531368484214431412842689938, 30.126175359519352365313190267348, 31.00622586279203101738747205914, 32.300249687306522621475576197

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