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Properties

Label 2-60-12.11-c1-0-5
Degree $2$
Conductor $60$
Sign $0.824 + 0.566i$
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  + (0.599 − 1.28i)2-s + (0.468 + 1.66i)3-s + (−1.28 − 1.53i)4-s − i·5-s + (2.41 + 0.400i)6-s + 0.936i·7-s + (−2.73 + 0.719i)8-s + (−2.56 + 1.56i)9-s + (−1.28 − 0.599i)10-s − 4.27·11-s + (1.96 − 2.85i)12-s + 3.12·13-s + (1.19 + 0.561i)14-s + (1.66 − 0.468i)15-s + (−0.719 + 3.93i)16-s + 2i·17-s + ⋯
L(s)  = 1  + (0.424 − 0.905i)2-s + (0.270 + 0.962i)3-s + (−0.640 − 0.768i)4-s − 0.447i·5-s + (0.986 + 0.163i)6-s + 0.353i·7-s + (−0.967 + 0.254i)8-s + (−0.853 + 0.520i)9-s + (−0.405 − 0.189i)10-s − 1.28·11-s + (0.566 − 0.824i)12-s + 0.866·13-s + (0.320 + 0.150i)14-s + (0.430 − 0.120i)15-s + (−0.179 + 0.983i)16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.824 + 0.566i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.824 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978736 - 0.303865i\)
\(L(\frac12)\) \(\approx\) \(0.978736 - 0.303865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.599 + 1.28i)T \)
3 \( 1 + (-0.468 - 1.66i)T \)
5 \( 1 + iT \)
good7 \( 1 - 0.936iT - 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 4.27iT - 19T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 + 0.936T + 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + 7.19T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 - 5.20iT - 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 - 9.06iT - 79T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−15.12721853998387560758954067671, −13.64744692494593608731451465108, −12.91423213666856429135643076153, −11.38050475721826584363927395409, −10.57857601017191980427561196748, −9.360767634609036809384467997366, −8.394129189816109509280775230956, −5.65432451686143326045612141125, −4.53852969408277104701046962527, −2.87913374273600121785135790634, 3.22694146055192446093530899592, 5.44369194765993277791341361341, 6.81823875920884828348016733970, 7.72522086622339893024831848000, 8.859104192445492659709432432835, 10.76511867283110920050760973735, 12.31425150702233224132889434805, 13.28666480287548796363482657370, 14.00710363071771586558179372761, 15.07545303584686628180404336943

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