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Properties

Label 1-61-61.31-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: $\chi_{61} (31, \cdot )$
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

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

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