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Properties

Label 2-60-60.59-c5-0-30
Degree $2$
Conductor $60$
Sign $-0.758 + 0.651i$
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  + (−3.28 − 4.60i)2-s + (−3.33 + 15.2i)3-s + (−10.3 + 30.2i)4-s + (−17.3 + 53.1i)5-s + (81.0 − 34.7i)6-s − 121.·7-s + (173. − 51.9i)8-s + (−220. − 101. i)9-s + (301. − 94.8i)10-s + 442.·11-s + (−426. − 258. i)12-s − 1.18e3i·13-s + (398. + 557. i)14-s + (−751. − 441. i)15-s + (−809. − 627. i)16-s − 1.03e3·17-s + ⋯
L(s)  = 1  + (−0.581 − 0.813i)2-s + (−0.213 + 0.976i)3-s + (−0.323 + 0.946i)4-s + (−0.310 + 0.950i)5-s + (0.919 − 0.394i)6-s − 0.934·7-s + (0.958 − 0.286i)8-s + (−0.908 − 0.417i)9-s + (0.953 − 0.299i)10-s + 1.10·11-s + (−0.855 − 0.518i)12-s − 1.94i·13-s + (0.543 + 0.760i)14-s + (−0.862 − 0.506i)15-s + (−0.790 − 0.612i)16-s − 0.866·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.758 + 0.651i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.758 + 0.651i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0505076 - 0.136338i\)
\(L(\frac12)\) \(\approx\) \(0.0505076 - 0.136338i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.28 + 4.60i)T \)
3 \( 1 + (3.33 - 15.2i)T \)
5 \( 1 + (17.3 - 53.1i)T \)
good7 \( 1 + 121.T + 1.68e4T^{2} \)
11 \( 1 - 442.T + 1.61e5T^{2} \)
13 \( 1 + 1.18e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 - 1.29e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.19e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.78e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.53e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.28e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.10e3iT - 1.15e8T^{2} \)
43 \( 1 - 194.T + 1.47e8T^{2} \)
47 \( 1 - 5.08e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.52e4T + 4.18e8T^{2} \)
59 \( 1 + 3.39e4T + 7.14e8T^{2} \)
61 \( 1 - 2.29e3T + 8.44e8T^{2} \)
67 \( 1 + 4.39e4T + 1.35e9T^{2} \)
71 \( 1 + 9.41e3T + 1.80e9T^{2} \)
73 \( 1 + 1.27e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.33e4iT - 3.07e9T^{2} \)
83 \( 1 - 3.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.10e5iT - 5.58e9T^{2} \)
97 \( 1 + 5.72e4iT - 8.58e9T^{2} \)
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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

−13.51405052045425177610241368886, −12.13194991941097222559230433519, −11.10688865746803579051733176862, −10.19401264667623792610785316759, −9.408114254081389588498441502170, −7.88686593615163293886427183392, −6.16542283886039591381982200524, −3.92357802213922815371597535285, −2.99829292808780255364993505753, −0.088857323314773531075996946074, 1.49004099909854744077984350019, 4.60409706040155026310623040370, 6.41267844594485512310897541620, 6.99774487656875988212127988810, 8.750616226089159473070105679298, 9.216731877139144932720505422764, 11.22600987407205399155923271090, 12.37142823500604554067535635104, 13.50249086175786459319131692923, 14.43509106819184245237960179952

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