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Properties

Label 1-60-60.59-r0-0-0
Degree $1$
Conductor $60$
Sign $1$
Analytic cond. $0.278638$
Root an. cond. $0.278638$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 11-s − 13-s + 17-s − 19-s − 23-s − 29-s − 31-s − 37-s − 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s − 89-s − 91-s − 97-s + ⋯
L(s)  = 1  + 7-s + 11-s − 13-s + 17-s − 19-s − 23-s − 29-s − 31-s − 37-s − 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s − 89-s − 91-s − 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(0.278638\)
Root analytic conductor: \(0.278638\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 60,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9618797159\)
\(L(\frac12)\) \(\approx\) \(0.9618797159\)
\(L(1)\) \(\approx\) \(1.065554320\)
\(L(1)\) \(\approx\) \(1.065554320\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−32.58262270514236334702277741913, −31.56412029546067390492273146917, −30.28370271838972914813392774493, −29.56153792303208874429096790028, −27.86125861653568212621604974582, −27.34677171069150831416030737867, −25.89354093744892243070839471337, −24.673927390561852209576229240424, −23.800425575409428011559961836279, −22.36432718487739382101850758381, −21.328431099039773488357789758822, −20.11799276794188069833039241236, −18.9285284080218204999053966311, −17.554960636106842748012592192039, −16.66706135687040887534297750505, −14.9070292412479863596844040316, −14.21207272889969396426157491711, −12.45813508004805044439217178808, −11.39941638239152605493477803422, −9.94252189246792938879975776348, −8.485164567234391764673881624397, −7.166280912138318471719556203672, −5.46144112265697581553500848609, −3.985806388187321605499007281285, −1.88060641687791880942488827805, 1.88060641687791880942488827805, 3.985806388187321605499007281285, 5.46144112265697581553500848609, 7.166280912138318471719556203672, 8.485164567234391764673881624397, 9.94252189246792938879975776348, 11.39941638239152605493477803422, 12.45813508004805044439217178808, 14.21207272889969396426157491711, 14.9070292412479863596844040316, 16.66706135687040887534297750505, 17.554960636106842748012592192039, 18.9285284080218204999053966311, 20.11799276794188069833039241236, 21.328431099039773488357789758822, 22.36432718487739382101850758381, 23.800425575409428011559961836279, 24.673927390561852209576229240424, 25.89354093744892243070839471337, 27.34677171069150831416030737867, 27.86125861653568212621604974582, 29.56153792303208874429096790028, 30.28370271838972914813392774493, 31.56412029546067390492273146917, 32.58262270514236334702277741913

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