Properties

Degree 1
Conductor 61
Sign $0.785 + 0.618i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.978 + 0.207i)2-s + (0.309 + 0.951i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (0.104 + 0.994i)6-s + (−0.669 − 0.743i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.104 − 0.994i)10-s − 11-s + (−0.104 + 0.994i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.913 − 0.406i)15-s + (0.669 + 0.743i)16-s + (−0.913 − 0.406i)17-s + ⋯
L(s,χ)  = 1  + (0.978 + 0.207i)2-s + (0.309 + 0.951i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (0.104 + 0.994i)6-s + (−0.669 − 0.743i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.104 − 0.994i)10-s − 11-s + (−0.104 + 0.994i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.913 − 0.406i)15-s + (0.669 + 0.743i)16-s + (−0.913 − 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.785 + 0.618i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.785 + 0.618i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.785 + 0.618i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (4, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (0:\ ),\ 0.785 + 0.618i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.414278594 + 0.4898206080i$
$L(\frac12,\chi)$  $\approx$  $1.414278594 + 0.4898206080i$
$L(\chi,1)$  $\approx$  1.555629262 + 0.4010186989i
$L(1,\chi)$  $\approx$  1.555629262 + 0.4010186989i

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

−31.96955237631432578467532039949, −31.25716882341293792682045621176, −30.38041527986491991769608257259, −29.3346888201728755444045513227, −28.63068061274698196744092337503, −26.51885989431183207824064796642, −25.37498071291631502925438576308, −24.58387797414741740204888607202, −23.1719100595719860166032918477, −22.56973847049369386869234223664, −21.25779779792881522813040213597, −19.76841334380398871602725311871, −18.98121879569901866943923779868, −17.83854977167380433796821435400, −15.67392963382198606725305508374, −14.87662667095666260907997069492, −13.54645564629807243767957919802, −12.66762753979436994562391233490, −11.504376790577464821186961043079, −10.07391254428021526681307949047, −7.88316718246811272320344391961, −6.677743376475393021203769952833, −5.58836308991348397017762219193, −3.205097676237261703086192638164, −2.39751250324684599921048316242, 2.81189508178981899589370068010, 4.35087092822791657307143467301, 5.11624504680452522285994609296, 7.02707471751810652480898881720, 8.65433739856954049247963756300, 10.128767483857445681718036559229, 11.53091061927997546087078918604, 13.06639772431607095568764088172, 13.88089980709336936416487019399, 15.4544387943471549077278417998, 16.2015435107319720450016118679, 17.069248602389740937054699752347, 19.65088449191733384295613246348, 20.4286475288511626521910487558, 21.332445687935360585669391860772, 22.47665470641909409760914522908, 23.60411545314197302495228264018, 24.660347088047661359952104556609, 26.0118642963064500999334953041, 26.76947483750982743232179300006, 28.57622214686134650174592190729, 29.15530675827763543315191326455, 31.04373476577759484538047737331, 31.6282657792352942620537577814, 32.714916389144374080975454291153

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