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Properties

Label 2-60-15.2-c3-0-5
Degree $2$
Conductor $60$
Sign $-0.427 + 0.903i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 − 5.05i)3-s + (−8.06 − 7.74i)5-s + (−3.75 − 3.75i)7-s + (−24.1 − 11.9i)9-s − 6.48i·11-s + (27.4 − 27.4i)13-s + (−48.7 + 31.6i)15-s + (55.8 − 55.8i)17-s − 10.4i·19-s + (−23.4 + 14.5i)21-s + (115. + 115. i)23-s + (4.99 + 124. i)25-s + (−89.3 + 108. i)27-s + 267.·29-s − 305.·31-s + ⋯
L(s)  = 1  + (0.228 − 0.973i)3-s + (−0.721 − 0.692i)5-s + (−0.202 − 0.202i)7-s + (−0.895 − 0.444i)9-s − 0.177i·11-s + (0.586 − 0.586i)13-s + (−0.839 + 0.543i)15-s + (0.796 − 0.796i)17-s − 0.126i·19-s + (−0.243 + 0.151i)21-s + (1.04 + 1.04i)23-s + (0.0399 + 0.999i)25-s + (−0.637 + 0.770i)27-s + 1.71·29-s − 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.427 + 0.903i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.427 + 0.903i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.645084 - 1.01904i\)
\(L(\frac12)\) \(\approx\) \(0.645084 - 1.01904i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.18 + 5.05i)T \)
5 \( 1 + (8.06 + 7.74i)T \)
good7 \( 1 + (3.75 + 3.75i)T + 343iT^{2} \)
11 \( 1 + 6.48iT - 1.33e3T^{2} \)
13 \( 1 + (-27.4 + 27.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (-55.8 + 55.8i)T - 4.91e3iT^{2} \)
19 \( 1 + 10.4iT - 6.85e3T^{2} \)
23 \( 1 + (-115. - 115. i)T + 1.21e4iT^{2} \)
29 \( 1 - 267.T + 2.43e4T^{2} \)
31 \( 1 + 305.T + 2.97e4T^{2} \)
37 \( 1 + (-187. - 187. i)T + 5.06e4iT^{2} \)
41 \( 1 + 374. iT - 6.89e4T^{2} \)
43 \( 1 + (-291. + 291. i)T - 7.95e4iT^{2} \)
47 \( 1 + (230. - 230. i)T - 1.03e5iT^{2} \)
53 \( 1 + (178. + 178. i)T + 1.48e5iT^{2} \)
59 \( 1 + 225.T + 2.05e5T^{2} \)
61 \( 1 + 315.T + 2.26e5T^{2} \)
67 \( 1 + (-203. - 203. i)T + 3.00e5iT^{2} \)
71 \( 1 - 161. iT - 3.57e5T^{2} \)
73 \( 1 + (-300. + 300. i)T - 3.89e5iT^{2} \)
79 \( 1 - 948. iT - 4.93e5T^{2} \)
83 \( 1 + (-553. - 553. i)T + 5.71e5iT^{2} \)
89 \( 1 + 577.T + 7.04e5T^{2} \)
97 \( 1 + (-84.9 - 84.9i)T + 9.12e5iT^{2} \)
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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

−14.00445748814179351262342868756, −13.03797444458528280641670961687, −12.16458957201486179832875011068, −11.08858910223900516381558790481, −9.230441058593903565202713180342, −8.112596349356525228743233821442, −7.10378401912048340079673789835, −5.41595817601067611836709729135, −3.34527370019963594306126767099, −0.889481576906678497085976615265, 3.09543962731563077319473840614, 4.42190080357317280540189800817, 6.26476227515677365298794401107, 7.927532147855946164369021542556, 9.142826676315940863106646836667, 10.46919374416374784467217305198, 11.26384455967736855335154488121, 12.60748717636432702679179523983, 14.30388331645715659460776322952, 14.87293318313190389767957149773

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