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Properties

Label 1-61-61.4-r0-0-0
Degree $1$
Conductor $61$
Sign $0.785 + 0.618i$
Analytic cond. $0.283282$
Root an. cond. $0.283282$
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  + (0.978 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.785 + 0.618i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.785 + 0.618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.414278594 + 0.4898206080i\)
\(L(\frac12)\) \(\approx\) \(1.414278594 + 0.4898206080i\)
\(L(1)\) \(\approx\) \(1.555629262 + 0.4010186989i\)
\(L(1)\) \(\approx\) \(1.555629262 + 0.4010186989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−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