Properties

Label 2-60-15.8-c5-0-0
Degree $2$
Conductor $60$
Sign $-0.942 + 0.334i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.61 + 14.8i)3-s + (−31.2 − 46.3i)5-s + (−91.3 + 91.3i)7-s + (−200. + 137. i)9-s − 99.2i·11-s + (−452. − 452. i)13-s + (545. − 679. i)15-s + (−1.26e3 − 1.26e3i)17-s + 912. i·19-s + (−1.78e3 − 939. i)21-s + (−1.74e3 + 1.74e3i)23-s + (−1.16e3 + 2.89e3i)25-s + (−2.97e3 − 2.35e3i)27-s − 299.·29-s + 2.59e3·31-s + ⋯
L(s)  = 1  + (0.295 + 0.955i)3-s + (−0.559 − 0.828i)5-s + (−0.705 + 0.705i)7-s + (−0.824 + 0.565i)9-s − 0.247i·11-s + (−0.742 − 0.742i)13-s + (0.625 − 0.780i)15-s + (−1.06 − 1.06i)17-s + 0.580i·19-s + (−0.882 − 0.464i)21-s + (−0.688 + 0.688i)23-s + (−0.373 + 0.927i)25-s + (−0.784 − 0.620i)27-s − 0.0661·29-s + 0.484·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.942 + 0.334i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.942 + 0.334i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0276879 - 0.160758i\)
\(L(\frac12)\) \(\approx\) \(0.0276879 - 0.160758i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.61 - 14.8i)T \)
5 \( 1 + (31.2 + 46.3i)T \)
good7 \( 1 + (91.3 - 91.3i)T - 1.68e4iT^{2} \)
11 \( 1 + 99.2iT - 1.61e5T^{2} \)
13 \( 1 + (452. + 452. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.26e3 + 1.26e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 912. iT - 2.47e6T^{2} \)
23 \( 1 + (1.74e3 - 1.74e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 299.T + 2.05e7T^{2} \)
31 \( 1 - 2.59e3T + 2.86e7T^{2} \)
37 \( 1 + (-5.91e3 + 5.91e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.94e4iT - 1.15e8T^{2} \)
43 \( 1 + (8.60e3 + 8.60e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.73e4 - 1.73e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.94e4 - 1.94e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 220.T + 7.14e8T^{2} \)
61 \( 1 - 4.39e4T + 8.44e8T^{2} \)
67 \( 1 + (3.86e4 - 3.86e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 6.18e4iT - 1.80e9T^{2} \)
73 \( 1 + (1.40e4 + 1.40e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 6.48e4iT - 3.07e9T^{2} \)
83 \( 1 + (5.90e3 - 5.90e3i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.25e5T + 5.58e9T^{2} \)
97 \( 1 + (7.06e4 - 7.06e4i)T - 8.58e9iT^{2} \)
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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

−15.02061338410048662103562070196, −13.59485708201374921239433099654, −12.40128932819558025187784404317, −11.34068092777403403836791484638, −9.817439928607655520517297423442, −9.020269044761666655498964030420, −7.83204054083171251550205568852, −5.71339477570731937788715477402, −4.45608992017659051052174447939, −2.89142982054699192583271354109, 0.06966336966313020284734351226, 2.36609088244197290938468359100, 3.98350554347688604335370165004, 6.51921145483580625195900505970, 7.11068562950670200202322189300, 8.451798794234376452569965416674, 10.02815271909952410404428741380, 11.31003366711749021945315943712, 12.42313931841766850789269364460, 13.48724167887929523472587645909

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