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Properties

Label 2-60-60.59-c5-0-45
Degree $2$
Conductor $60$
Sign $0.885 + 0.463i$
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  + (5.44 + 1.52i)2-s + (11.2 − 10.8i)3-s + (27.3 + 16.6i)4-s + (−42.6 − 36.1i)5-s + (77.5 − 41.8i)6-s + 128.·7-s + (123. + 132. i)8-s + (8.22 − 242. i)9-s + (−176. − 262. i)10-s + 306.·11-s + (486. − 109. i)12-s − 408. i·13-s + (702. + 196. i)14-s + (−869. + 55.9i)15-s + (470. + 909. i)16-s − 1.24e3·17-s + ⋯
L(s)  = 1  + (0.962 + 0.269i)2-s + (0.718 − 0.695i)3-s + (0.854 + 0.519i)4-s + (−0.762 − 0.647i)5-s + (0.879 − 0.475i)6-s + 0.994·7-s + (0.682 + 0.731i)8-s + (0.0338 − 0.999i)9-s + (−0.559 − 0.829i)10-s + 0.764·11-s + (0.975 − 0.219i)12-s − 0.670i·13-s + (0.957 + 0.268i)14-s + (−0.997 + 0.0642i)15-s + (0.459 + 0.888i)16-s − 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.463i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.885 + 0.463i$
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: \((2,\ 60,\ (\ :5/2),\ 0.885 + 0.463i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.58439 - 0.881827i\)
\(L(\frac12)\) \(\approx\) \(3.58439 - 0.881827i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.44 - 1.52i)T \)
3 \( 1 + (-11.2 + 10.8i)T \)
5 \( 1 + (42.6 + 36.1i)T \)
good7 \( 1 - 128.T + 1.68e4T^{2} \)
11 \( 1 - 306.T + 1.61e5T^{2} \)
13 \( 1 + 408. iT - 3.71e5T^{2} \)
17 \( 1 + 1.24e3T + 1.41e6T^{2} \)
19 \( 1 - 1.54e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.93e3iT - 6.43e6T^{2} \)
29 \( 1 - 7.47e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.18e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.16e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.56e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.90e4T + 1.47e8T^{2} \)
47 \( 1 - 1.15e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.36e4T + 4.18e8T^{2} \)
59 \( 1 + 1.93e4T + 7.14e8T^{2} \)
61 \( 1 - 3.41e4T + 8.44e8T^{2} \)
67 \( 1 - 4.93e4T + 1.35e9T^{2} \)
71 \( 1 - 7.17e4T + 1.80e9T^{2} \)
73 \( 1 + 8.10e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.16e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.59e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.76e3iT - 5.58e9T^{2} \)
97 \( 1 + 1.23e5iT - 8.58e9T^{2} \)
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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

−14.15526123422798653718978823977, −12.82788916430448147050398702955, −12.17929337977917268199647622977, −11.05593819894112633489489856854, −8.689897192048326772835602773495, −7.933226975935964125870958985321, −6.69824965645035589031529727345, −4.89609117575485525634785056908, −3.53954202488901697727933976540, −1.58864385441211429887714471532, 2.22185796619213418774381689062, 3.82855479429118597551882655045, 4.72569890937985301713614803529, 6.76519456345321158711458866724, 8.113924099019876760974184857626, 9.693890019637867839653782499171, 11.31857217244900451237968374628, 11.42891583533673369887463026817, 13.41406759801338150380592960632, 14.27350670844241362803390185860

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