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Properties

Label 1-61-61.46-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: $\chi_{61} (46, \cdot )$
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

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

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