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Properties

Label 2-60-5.4-c3-0-0
Degree $2$
Conductor $60$
Sign $-0.447 - 0.894i$
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  + 3i·3-s + (−10 + 5i)5-s + 22i·7-s − 9·9-s − 14·11-s + 30i·13-s + (−15 − 30i)15-s + 62i·17-s + 120·19-s − 66·21-s − 188i·23-s + (75 − 100i)25-s − 27i·27-s − 96·29-s + 184·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.894 + 0.447i)5-s + 1.18i·7-s − 0.333·9-s − 0.383·11-s + 0.640i·13-s + (−0.258 − 0.516i)15-s + 0.884i·17-s + 1.44·19-s − 0.685·21-s − 1.70i·23-s + (0.599 − 0.800i)25-s − 0.192i·27-s − 0.614·29-s + 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.539763 + 0.873355i\)
\(L(\frac12)\) \(\approx\) \(0.539763 + 0.873355i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
5 \( 1 + (10 - 5i)T \)
good7 \( 1 - 22iT - 343T^{2} \)
11 \( 1 + 14T + 1.33e3T^{2} \)
13 \( 1 - 30iT - 2.19e3T^{2} \)
17 \( 1 - 62iT - 4.91e3T^{2} \)
19 \( 1 - 120T + 6.85e3T^{2} \)
23 \( 1 + 188iT - 1.21e4T^{2} \)
29 \( 1 + 96T + 2.43e4T^{2} \)
31 \( 1 - 184T + 2.97e4T^{2} \)
37 \( 1 - 406iT - 5.06e4T^{2} \)
41 \( 1 - 130T + 6.89e4T^{2} \)
43 \( 1 + 148iT - 7.95e4T^{2} \)
47 \( 1 - 448iT - 1.03e5T^{2} \)
53 \( 1 - 414iT - 1.48e5T^{2} \)
59 \( 1 + 266T + 2.05e5T^{2} \)
61 \( 1 + 838T + 2.26e5T^{2} \)
67 \( 1 - 248iT - 3.00e5T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + 484iT - 3.89e5T^{2} \)
79 \( 1 - 48T + 4.93e5T^{2} \)
83 \( 1 + 548iT - 5.71e5T^{2} \)
89 \( 1 - 650T + 7.04e5T^{2} \)
97 \( 1 + 1.81e3iT - 9.12e5T^{2} \)
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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

−15.18164340988960176128298056831, −14.11827393954947031647812105566, −12.40581078479412414450720146950, −11.59579945449408732215424199311, −10.43190617814385633843545324693, −9.045254492107358451481695548043, −7.936124825875806050052418129540, −6.23463771150623756282406870867, −4.61418603474678512847924090039, −2.92707269021384434319322824921, 0.75279567343431465149817209523, 3.52206508037291214315350357922, 5.23962165145979374907026878727, 7.30238410986779681023005762665, 7.81023775929059307732008724150, 9.508482528601366239430836201492, 11.00645567177374432889388036053, 11.96626871538909193857831557204, 13.19361116250847924129284430140, 13.96003666864334054483352860161

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