Properties

Label 1-61-61.33-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.180 + 0.983i$
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  + (−0.587 + 0.809i)2-s + (0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.951 + 0.309i)6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)10-s + i·11-s + (0.309 − 0.951i)12-s + 13-s − 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.309i)17-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.951 + 0.309i)6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)10-s + i·11-s + (0.309 − 0.951i)12-s + 13-s − 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.180 + 0.983i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9605985573 + 1.153378356i\)
\(L(\frac12)\) \(\approx\) \(0.9605985573 + 1.153378356i\)
\(L(1)\) \(\approx\) \(0.9200223296 + 0.5781922535i\)
\(L(1)\) \(\approx\) \(0.9200223296 + 0.5781922535i\)

Euler product

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

−31.34964665157505199977098645748, −30.62808570019724883861211334227, −29.93686075299808910059369538205, −28.92169828787039271890730988907, −27.09076740517131312046037647581, −26.66685714332376400845991516118, −25.63883745997412805998444966424, −24.160990264969102619051694928611, −22.863346187071786667263441728587, −21.40688330015312300555494717696, −20.35121483122237218473046267202, −19.362717146104247032870986110326, −18.45073037555951310978892274512, −17.54499034557236254004745682975, −15.72377923283646963620346143481, −14.00790773858666180632774626140, −13.376596196259625462715244510972, −11.516317920767679075835170915970, −10.76027635678723243017446626444, −9.07032534296070579973307971462, −7.91432422607200669570060720450, −6.87883703929164044981871516220, −3.88330498078853627410070230971, −2.79016874802023989415264285272, −1.02089241963732251659262439122, 1.74142487925873831305144242758, 4.31085857840227942807945091736, 5.447468518343120997306431323465, 7.54805750889493578235339741771, 8.69470352364436128045047708395, 9.29111007670792779188675820251, 10.94605750845168363650931863189, 12.876718213408038383256722991194, 14.3677449273071475913716385242, 15.435517991633923295077054651327, 16.11601436715816141859167269713, 17.55683869120718412623572777191, 18.80589084959483670305537086829, 20.11615183067527987052874936257, 20.8898896662156903189261491836, 22.56331448247042394800774784869, 24.059111791889619539898091142893, 24.94610374349597626512764235215, 25.74300202076205741852007013480, 27.04838662388097532095509397847, 27.916468624749715065414557275003, 28.58625811340366211627066236655, 31.05852231673508282283044768751, 31.416082783097034457662541227210, 32.99234239165709658783276095191

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