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Properties

Label 1-61-61.58-r0-0-0
Degree $1$
Conductor $61$
Sign $0.966 + 0.258i$
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  + (−0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)10-s + 11-s + (0.309 − 0.951i)12-s + 13-s + 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)10-s + 11-s + (0.309 − 0.951i)12-s + 13-s + 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.966 + 0.258i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.966 + 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4843623672 + 0.06369121931i\)
\(L(\frac12)\) \(\approx\) \(0.4843623672 + 0.06369121931i\)
\(L(1)\) \(\approx\) \(0.5885949034 - 0.03214788653i\)
\(L(1)\) \(\approx\) \(0.5885949034 - 0.03214788653i\)

Euler product

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

−32.825417642657353876978323999437, −32.066602989128792604264697321730, −29.678961438500610490274205758, −28.9002728691096926426159996620, −27.90589375252955716935945720643, −27.16222423798941228591362008525, −25.83632732245356890431732053848, −24.82662268696863475076028139614, −23.50888662421658071887326198652, −22.69088378802478934898026818053, −20.96201748835704013743554049686, −19.99350828200082411558579673684, −18.50398383645775779891655945731, −17.10542061626726984847843140087, −16.59433990255974720987961210135, −15.70784294112602011634759146961, −14.02530641143294031239006674721, −12.34198066066672024699925315857, −10.820723598531887699996993294869, −9.68685677256375033403236106726, −8.759180180698813872004108362587, −6.76179524024987167885184412042, −5.76696763212073967115518834137, −4.21985823830064048161307330507, −0.96987348686194981660967633127, 1.75407643977292948436363372247, 3.44232421599187528284126097696, 6.15474778656880661568942829320, 6.94731631960022808262803759181, 8.72376421244395568298954388918, 10.20885407646901477354803847252, 11.208778132532111743120423451705, 12.31177636482575255775174866244, 13.5034952500916101846199793114, 15.52493212387547453354180049277, 16.91323927278695187263433560886, 17.838736945197663772717649368315, 18.906070079968717474201214581884, 19.493940418270287929073822331468, 21.564445570927880437898414070111, 22.1758350830280417628706793675, 23.397506459795835542278261688442, 25.27268771584705284309909805430, 25.73308665124085737280189400929, 27.339222370363579133651103126140, 28.28005797206906193293098706729, 29.228839146447341398791077554784, 30.13380977217312287142524451546, 30.83460581035307406756849862231, 32.81153362825632703660751128939

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