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Properties

Label 2-60-60.59-c5-0-51
Degree $2$
Conductor $60$
Sign $-0.674 + 0.738i$
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  + (1.03 + 5.56i)2-s + (13.3 − 8.09i)3-s + (−29.8 + 11.5i)4-s + (−55.7 − 4.70i)5-s + (58.8 + 65.6i)6-s − 92.5·7-s + (−95.0 − 154. i)8-s + (111. − 215. i)9-s + (−31.6 − 314. i)10-s − 666.·11-s + (−304. + 395. i)12-s + 717. i·13-s + (−95.9 − 514. i)14-s + (−779. + 388. i)15-s + (758. − 688. i)16-s − 525.·17-s + ⋯
L(s)  = 1  + (0.183 + 0.983i)2-s + (0.854 − 0.519i)3-s + (−0.932 + 0.360i)4-s + (−0.996 − 0.0840i)5-s + (0.667 + 0.744i)6-s − 0.713·7-s + (−0.525 − 0.850i)8-s + (0.460 − 0.887i)9-s + (−0.0999 − 0.994i)10-s − 1.66·11-s + (−0.609 + 0.792i)12-s + 1.17i·13-s + (−0.130 − 0.701i)14-s + (−0.895 + 0.445i)15-s + (0.740 − 0.672i)16-s − 0.440·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.674 + 0.738i$
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: \((2,\ 60,\ (\ :5/2),\ -0.674 + 0.738i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0137847 - 0.0312529i\)
\(L(\frac12)\) \(\approx\) \(0.0137847 - 0.0312529i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 5.56i)T \)
3 \( 1 + (-13.3 + 8.09i)T \)
5 \( 1 + (55.7 + 4.70i)T \)
good7 \( 1 + 92.5T + 1.68e4T^{2} \)
11 \( 1 + 666.T + 1.61e5T^{2} \)
13 \( 1 - 717. iT - 3.71e5T^{2} \)
17 \( 1 + 525.T + 1.41e6T^{2} \)
19 \( 1 + 1.76e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.08e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.49e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.60e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.28e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.26e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.45e4T + 1.47e8T^{2} \)
47 \( 1 + 5.57e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.14e4T + 4.18e8T^{2} \)
59 \( 1 + 8.48e3T + 7.14e8T^{2} \)
61 \( 1 - 6.58e3T + 8.44e8T^{2} \)
67 \( 1 + 3.90e4T + 1.35e9T^{2} \)
71 \( 1 + 2.48e3T + 1.80e9T^{2} \)
73 \( 1 + 2.20e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.08e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.73e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.04e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.59e5iT - 8.58e9T^{2} \)
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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

−13.50503100696936348303830075431, −13.12892939081438149073558548890, −11.74408949366244398384696492239, −9.693762564609414457413188685167, −8.528124951701704453114881555228, −7.57739826418214946840557817914, −6.60597544445390831860679295029, −4.59353432154990160742502264830, −3.08284403920643225952852813417, −0.01323342374551460981070083479, 2.70010794304660991295039496339, 3.68847386174094693883217959313, 5.17953697749561824513236394979, 7.75887644464787470222814733024, 8.700230984306639797989153515733, 10.24891664147898370695364673302, 10.71433436818701337097790813259, 12.50320271294073606001962183461, 13.09502918405561628670353136849, 14.45286964246511562582889076721

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