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Properties

Label 1-61-61.29-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.443 + 0.896i$
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  + (−0.866 − 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s − i·8-s + 9-s + (−0.866 + 0.5i)10-s − i·11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s − i·8-s + 9-s + (−0.866 + 0.5i)10-s − i·11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.443 + 0.896i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.443 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01875464394 + 0.03020146486i\)
\(L(\frac12)\) \(\approx\) \(0.01875464394 + 0.03020146486i\)
\(L(1)\) \(\approx\) \(0.3944942614 - 0.1333700050i\)
\(L(1)\) \(\approx\) \(0.3944942614 - 0.1333700050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 - T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−32.4653491707956370705504668269, −30.44111246410599790531030791957, −29.22813067525953194627858148323, −28.62893196299466740998638259476, −27.48199995002936883019572306976, −26.33064039644208059712801556544, −25.34770860762089658593680476075, −24.25033799314656208241554655526, −22.653136465254671922989180674686, −22.289754544806981519297653882549, −20.302609555942451392771364433248, −18.83771490860282680196582404974, −17.99605928564159416107878907484, −17.19214481460316890777343851256, −15.785165862601395493146781986668, −14.96977493017815834316803367474, −13.02399791240675099478568778063, −11.462343050335716012005244109364, −10.23764869542733081511735755358, −9.45927179783249305942682059588, −7.25955841717414537161300344354, −6.45956370294922425901554069115, −5.22150529662467815418013345773, −2.37124496261525744357359601999, −0.02824507222864252904560514969, 1.51461807475693791303864337812, 3.896156376470791939743552222595, 5.821304398383507838824773178112, 7.16945517212981212990423067084, 8.99295392709322618558137109325, 10.00108579377901186252619960474, 11.22717126592770333152979987524, 12.444818758401182469202420792002, 13.41631065224209745687673702719, 16.13452055595458308398680724042, 16.60201958331909540125883859105, 17.57177783533822779998541492874, 18.87623864473610267375674368562, 19.9755733976472869952003986932, 21.37853973312335779031222551474, 22.10250960679395191786576754509, 23.78183713219788411991688720549, 24.800192489961122719676579318670, 26.251650804655206400585906911679, 27.232768877937329314965114566574, 28.43902412256740702274914648856, 29.21669882143501134629887927221, 29.62266814692616569204243775351, 31.468481493647434783791030234257, 32.87631921725932070481217725066

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