Properties

Degree 1
Conductor $ 5 \cdot 11 $
Sign $0.591 - 0.805i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $0.591 - 0.805i$
motivic weight  =  \(0\)
character  :  $\chi_{55} (18, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 55,\ (0:\ ),\ 0.591 - 0.805i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6015252823 - 0.3045210290i$
$L(\frac12,\chi)$  $\approx$  $0.6015252823 - 0.3045210290i$
$L(\chi,1)$  $\approx$  0.7552325654 - 0.2037787789i
$L(1,\chi)$  $\approx$  0.7552325654 - 0.2037787789i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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