Properties

Label 1-55-55.18-r0-0-0
Degree $1$
Conductor $55$
Sign $0.591 - 0.805i$
Analytic cond. $0.255418$
Root an. cond. $0.255418$
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  + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s i·12-s + (0.951 − 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s + (−0.809 − 0.587i)19-s − 21-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s i·12-s + (0.951 − 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s + (−0.809 − 0.587i)19-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.591 - 0.805i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ 0.591 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6015252823 - 0.3045210290i\)
\(L(\frac12)\) \(\approx\) \(0.6015252823 - 0.3045210290i\)
\(L(1)\) \(\approx\) \(0.7552325654 - 0.2037787789i\)
\(L(1)\) \(\approx\) \(0.7552325654 - 0.2037787789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 - T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−33.44852619504023484507361318603, −32.11569064509118051929692827918, −31.07585511878213084430867887726, −29.80836243059233981933081153068, −28.341831107605274316281076146321, −27.811721470520823404649369355532, −26.46571966627762439938134028428, −25.6739408309677949453018913172, −24.80424036285825608461098153398, −22.69433510054855268837378208897, −21.35044324312816573622487261805, −20.66597244128475876509736378019, −19.2672383282265299122352488859, −18.522846736946335125762241788228, −16.71078093837538622141847516835, −15.930888292423753501853459904326, −14.71796568623270028654463746346, −12.85792596706352835014149407280, −11.34644918886600785666300469893, −10.04452907282281449538539882809, −9.06865526846450957342309569478, −8.01963658548738103231823015171, −6.06655135828151376706037216774, −3.76133618555819924792020344552, −2.36600792071184758719429553955, 1.30014287585615686949992498315, 3.25200110821350319556228540660, 6.11180428462399123348636591563, 7.24138714660632625498829362591, 8.353630698535998399571659245460, 9.650089708701719580591370226, 11.05530548167924159866682298584, 12.760094011726534490809635024716, 14.043317159044108590604531462578, 15.39885402215489735414780833778, 16.77721799707679171390028764390, 17.9035879911143490844142356776, 19.05731816597585972340950116661, 19.85542858096311378255085093057, 20.937808056279029779046730180934, 23.24850084874625411722075256305, 23.94058092164055394110409502357, 25.54727012593731484441535212437, 25.78044284523823977756604392479, 27.15262531221857430921273879887, 28.458558931905597961114649001837, 29.66514808738799584521884751837, 30.29687496527560287287722357187, 32.03591944251538399274657685461, 33.00651556452398433063924318788

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