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Properties

Label 1-61-61.25-r0-0-0
Degree $1$
Conductor $61$
Sign $0.961 - 0.274i$
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  + (−0.104 − 0.994i)2-s + (−0.809 + 0.587i)3-s + (−0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (0.669 + 0.743i)6-s + (0.913 − 0.406i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.669 − 0.743i)10-s + 11-s + (0.669 − 0.743i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.978 − 0.207i)15-s + (0.913 − 0.406i)16-s + (−0.978 + 0.207i)17-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.809 + 0.587i)3-s + (−0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (0.669 + 0.743i)6-s + (0.913 − 0.406i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.669 − 0.743i)10-s + 11-s + (0.669 − 0.743i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.978 − 0.207i)15-s + (0.913 − 0.406i)16-s + (−0.978 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.961 - 0.274i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.961 - 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7403473391 - 0.1037281636i\)
\(L(\frac12)\) \(\approx\) \(0.7403473391 - 0.1037281636i\)
\(L(1)\) \(\approx\) \(0.8367636724 - 0.1426376074i\)
\(L(1)\) \(\approx\) \(0.8367636724 - 0.1426376074i\)

Euler product

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

−33.06239702054554814907041400352, −31.66164543293591261485417287777, −30.421388487456614107055802053218, −29.10813281475016799510333225314, −27.92111794838886590323862598847, −27.29939164566381526357150588566, −25.42819827107497880139848365837, −24.49074252538584279636860362309, −24.17367867342288695869098710084, −22.537153700007019069784871760294, −21.77826605690586825515381583790, −19.915491022211390972090101578, −18.236708038976318981568744329886, −17.53465830212968069809878980424, −16.79107988430018248875624066587, −15.35061858358446490891382464594, −13.89979081411186377061376908647, −12.86438940111460314070003214442, −11.49691107086308419841869489106, −9.6368935751665988726004862596, −8.34133137275738271875747332254, −6.9789804410425200575336501816, −5.590975763561288996581448373627, −4.83632698834199109446534768184, −1.403106250135808605456712204177, 1.77323360668691005286072937344, 3.82810473371510632834363942522, 5.07156193808838368761729162265, 6.804558475609646795966515930318, 9.02737489309637951544668794533, 10.17133865803463520874706008972, 11.14172998837427743165559705488, 12.00323838411098243396881600685, 13.84029895579861697552286653675, 14.75634663551233197420229100293, 16.89675277241131947651042729327, 17.60024447397073947177343072643, 18.64687388430699458772902903325, 20.22724754040496741180165866386, 21.36759066415015833587723861108, 22.092291441354158534606082781211, 22.99202663018335828304633625226, 24.475947554351933311458514549468, 26.61478745664405525181879423130, 26.856230210613579461794539758921, 28.24379436326196401875946982942, 29.14569203932420292810624467711, 30.08348613047981199827967053185, 31.03034517310895615527179769351, 32.69273104435051775120716007064

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