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Properties

Label 2-60-60.59-c3-0-0
Degree $2$
Conductor $60$
Sign $-0.952 + 0.304i$
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  + 2.82i·2-s + (−1.58 − 4.94i)3-s − 8.00·4-s + 11.1i·5-s + (14.0 − 4.47i)6-s − 34.7·7-s − 22.6i·8-s + (−22 + 15.6i)9-s − 31.6·10-s + (12.6 + 39.5i)12-s − 98.3i·14-s + (55.3 − 17.6i)15-s + 64.0·16-s + (−44.2 − 62.2i)18-s − 89.4i·20-s + (55.0 + 172. i)21-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.304 − 0.952i)3-s − 1.00·4-s + 0.999i·5-s + (0.952 − 0.304i)6-s − 1.87·7-s − 1.00i·8-s + (−0.814 + 0.579i)9-s − 1.00·10-s + (0.304 + 0.952i)12-s − 1.87i·14-s + (0.952 − 0.304i)15-s + 1.00·16-s + (−0.579 − 0.814i)18-s − 1.00i·20-s + (0.571 + 1.78i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.952 + 0.304i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.952 + 0.304i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0386526 - 0.248027i\)
\(L(\frac12)\) \(\approx\) \(0.0386526 - 0.248027i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + (1.58 + 4.94i)T \)
5 \( 1 - 11.1iT \)
good7 \( 1 + 34.7T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 - 94.7iT - 1.21e4T^{2} \)
29 \( 1 - 62.6iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 460. iT - 6.89e4T^{2} \)
43 \( 1 + 376.T + 7.95e4T^{2} \)
47 \( 1 - 425. iT - 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 952T + 2.26e5T^{2} \)
67 \( 1 + 774.T + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 108. iT - 5.71e5T^{2} \)
89 \( 1 - 948. iT - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−15.29320136083900605065468630325, −13.96009563294127242885971370082, −13.23394302862210095716747718959, −12.19244037418500159836967494953, −10.45065916963449620267226889324, −9.215609094872675604269319369777, −7.51446563720878588400609749975, −6.67508787827811137153131290096, −5.85500024695023818099067164303, −3.27784160516632898521443723557, 0.17666639000619821441175926180, 3.22782123357478675252932884824, 4.55132469486312377704126186131, 6.02965178541942142799017726881, 8.662225897606281002936080979194, 9.584201349598170256215612189602, 10.27013998681000470387754088628, 11.77436816806983265864793781469, 12.68796288633509874992903949695, 13.56845078342958141928655765263

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