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Properties

Label 1-61-61.2-r1-0-0
Degree $1$
Conductor $61$
Sign $0.595 + 0.803i$
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  + (0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (0.743 + 0.669i)6-s + (0.406 − 0.913i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.743 + 0.669i)10-s + i·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 + (0.207 − 0.978i)17-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (0.743 + 0.669i)6-s + (0.406 − 0.913i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.743 + 0.669i)10-s + i·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 + (0.207 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ 0.595 + 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.056810096 + 1.539653966i\)
\(L(\frac12)\) \(\approx\) \(3.056810096 + 1.539653966i\)
\(L(1)\) \(\approx\) \(2.162905402 + 0.7015393805i\)
\(L(1)\) \(\approx\) \(2.162905402 + 0.7015393805i\)

Euler product

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

−31.65287617018018287103706001491, −31.387139625133679555110025081650, −30.1979964111145380945478400785, −29.06261157747973655526016901201, −27.87250286391893554722090148855, −26.25989708780992645354555022883, −24.88260380311178063627568915209, −24.236657236245812379972182186409, −23.48389846659874297689908343806, −21.62031426184870613042389800336, −20.97545071142486651160250813466, −19.51298577484744484874200081533, −19.00648500903626035364585623565, −16.85231316539044042165131792240, −15.4238360577976676769677004158, −14.61536802298279399685817046725, −13.25577417084038043208536545165, −12.32893324401736749264064589912, −11.31780702166872876133436341775, −8.98564405574397241905095369722, −7.91699955261102630072061639319, −6.32731732881859186548327858266, −4.70142537379903443987494111653, −3.21825284006014094219871293014, −1.6392777144862769547634036168, 2.50492478939798634306244978568, 3.82008907179530277404721430885, 4.84182423162079509317457083765, 7.105127777473279405411257280035, 7.86375865433077037807682358532, 10.127405964020545874089439518721, 11.07758745729257320840860973556, 12.68448383366886363246489862068, 14.09967608395336902810296034606, 14.86245349736555501342009724897, 15.70454318366295968545515403439, 17.18081477538689481751735778074, 19.2026147286988288693072449225, 20.27648856484270043450312228492, 20.96085046906978597916947012826, 22.54969297999003630692772723910, 23.07240337086087907896122197701, 24.60022195064083813877019340647, 25.6718958272306965271604174096, 26.72489355221196817206648440708, 27.719064875578742372841281421700, 29.73056870068078452650451468793, 30.44809158626540686023945285589, 31.35557117768247956236681792947, 32.28492932795276832255978359097

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