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Properties

Label 2-60-60.59-c1-0-3
Degree $2$
Conductor $60$
Sign $0.968 + 0.250i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 0.866i)2-s − 1.73i·3-s + (0.500 − 1.93i)4-s + 2.23·5-s + (1.49 + 1.93i)6-s + (1.11 + 2.59i)8-s − 2.99·9-s + (−2.50 + 1.93i)10-s + (−3.35 − 0.866i)12-s − 3.87i·15-s + (−3.5 − 1.93i)16-s − 4.47·17-s + (3.35 − 2.59i)18-s + 7.74i·19-s + (1.11 − 4.33i)20-s + ⋯
L(s)  = 1  + (−0.790 + 0.612i)2-s − 0.999i·3-s + (0.250 − 0.968i)4-s + 0.999·5-s + (0.612 + 0.790i)6-s + (0.395 + 0.918i)8-s − 0.999·9-s + (−0.790 + 0.612i)10-s + (−0.968 − 0.250i)12-s − 1.00i·15-s + (−0.875 − 0.484i)16-s − 1.08·17-s + (0.790 − 0.612i)18-s + 1.77i·19-s + (0.250 − 0.968i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681006 - 0.0864991i\)
\(L(\frac12)\) \(\approx\) \(0.681006 - 0.0864991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.866i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 - 2.23T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 7.74iT - 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.98360755115807787929832308885, −13.99849049763453243297422380529, −13.10606584605855036134029401261, −11.57586224837614541281893136089, −10.23193054429604855466787590971, −9.031933621810800486731780558847, −7.84751197970826154652720172425, −6.55054750427678082702588075641, −5.61553303794249346366429241746, −1.91753482664858860475141963054, 2.71428527524336496712653239371, 4.71240468314306366469445297338, 6.64970921095770118350862487943, 8.680362817376058769151202987593, 9.384786331947788898180251109218, 10.50611649443584399182254907936, 11.26835583302356152021837947426, 12.83628333467048707080950419358, 13.97280879889808141111639535322, 15.42118884050048069651714128305

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