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Properties

Label 1-61-61.23-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.397 - 0.917i$
Analytic cond. $6.55536$
Root an. cond. $6.55536$
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  + (0.951 − 0.309i)2-s + (−0.309 − 0.951i)3-s + (0.809 − 0.587i)4-s + (0.809 − 0.587i)5-s + (−0.587 − 0.809i)6-s + (−0.951 − 0.309i)7-s + (0.587 − 0.809i)8-s + (−0.809 + 0.587i)9-s + (0.587 − 0.809i)10-s + i·11-s + (−0.809 − 0.587i)12-s + 13-s − 14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.587 − 0.809i)17-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (−0.309 − 0.951i)3-s + (0.809 − 0.587i)4-s + (0.809 − 0.587i)5-s + (−0.587 − 0.809i)6-s + (−0.951 − 0.309i)7-s + (0.587 − 0.809i)8-s + (−0.809 + 0.587i)9-s + (0.587 − 0.809i)10-s + i·11-s + (−0.809 − 0.587i)12-s + 13-s − 14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.587 − 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.397 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.416901907 - 2.157174916i\)
\(L(\frac12)\) \(\approx\) \(1.416901907 - 2.157174916i\)
\(L(1)\) \(\approx\) \(1.432637886 - 1.058363532i\)
\(L(1)\) \(\approx\) \(1.432637886 - 1.058363532i\)

Euler product

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

−32.63765576654705878335826773677, −31.91132097407370194315365113855, −30.50839837962107128602520207908, −29.25662219981129757666141814137, −28.5595022690458694496159917135, −26.637625820094408315257828128064, −25.92543487722380333080898841606, −24.86588083513290535118662723881, −23.20502698218727570793814179847, −22.4032377830681974733261622698, −21.5613565636714571728374813061, −20.740418677231609676993712299024, −18.974846534977507598197525611914, −17.22062159659972501107588154478, −16.23127334153621010001735387256, −15.241922093449224174811347569663, −14.05102213175066794608530441616, −12.91810541376932090533074348256, −11.26753202697077773590010631892, −10.28945506660223942807813167536, −8.669534580833369873362368655525, −6.24073180510781321064146842804, −5.93098703940762554526426287718, −3.9848636548020565143967713568, −2.814643701804694881417634084539, 1.17414380786346683750904152519, 2.70221827363724023279031521432, 4.78505937762207406698146849926, 6.156905662686844648367245850397, 7.06742756349191909859990650391, 9.29846908226799023081758783462, 10.83041215852980022191030869068, 12.27685363050011737922318099453, 13.21363893296708706594731781693, 13.75090027914150717822807983210, 15.61939732918869848744806734860, 16.91417718098172373236187242911, 18.2084847461744993424308223140, 19.67561931541017013704684602822, 20.48551406384507967052627650438, 21.950978767078270789823480218, 23.00703068381589765268985266674, 23.8066727764079101556213009130, 25.171313252495831720951536172016, 25.62016249088834740063527683540, 28.231554845026023947513998982490, 28.83081024229257449323399389058, 29.73257898270337532085946438350, 30.68787854633578774583077929942, 31.84173622268780151106719903422

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