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Properties

Label 2-60-60.47-c2-0-1
Degree $2$
Conductor $60$
Sign $-0.712 - 0.701i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 + 0.141i)2-s + (−2.06 − 2.17i)3-s + (3.95 − 0.565i)4-s + (−3.07 + 3.94i)5-s + (4.43 + 4.04i)6-s + (−5.18 + 5.18i)7-s + (−7.81 + 1.68i)8-s + (−0.459 + 8.98i)9-s + (5.57 − 8.29i)10-s − 7.14·11-s + (−9.41 − 7.44i)12-s + (−7.93 + 7.93i)13-s + (9.61 − 11.0i)14-s + (14.9 − 1.45i)15-s + (15.3 − 4.47i)16-s + (16.5 − 16.5i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0708i)2-s + (−0.688 − 0.724i)3-s + (0.989 − 0.141i)4-s + (−0.615 + 0.788i)5-s + (0.738 + 0.674i)6-s + (−0.741 + 0.741i)7-s + (−0.977 + 0.211i)8-s + (−0.0510 + 0.998i)9-s + (0.557 − 0.829i)10-s − 0.649·11-s + (−0.784 − 0.620i)12-s + (−0.610 + 0.610i)13-s + (0.686 − 0.791i)14-s + (0.995 − 0.0970i)15-s + (0.960 − 0.279i)16-s + (0.975 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0772723 + 0.188491i\)
\(L(\frac12)\) \(\approx\) \(0.0772723 + 0.188491i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.141i)T \)
3 \( 1 + (2.06 + 2.17i)T \)
5 \( 1 + (3.07 - 3.94i)T \)
good7 \( 1 + (5.18 - 5.18i)T - 49iT^{2} \)
11 \( 1 + 7.14T + 121T^{2} \)
13 \( 1 + (7.93 - 7.93i)T - 169iT^{2} \)
17 \( 1 + (-16.5 + 16.5i)T - 289iT^{2} \)
19 \( 1 + 12.1T + 361T^{2} \)
23 \( 1 + (11.0 - 11.0i)T - 529iT^{2} \)
29 \( 1 + 26.1T + 841T^{2} \)
31 \( 1 - 8.74iT - 961T^{2} \)
37 \( 1 + (26.7 + 26.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 35.4iT - 1.68e3T^{2} \)
43 \( 1 + (-24.6 - 24.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-58.6 - 58.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-20.4 - 20.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 59.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.42T + 3.72e3T^{2} \)
67 \( 1 + (35.8 - 35.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 46.2T + 5.04e3T^{2} \)
73 \( 1 + (-10.6 + 10.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 68.3T + 6.24e3T^{2} \)
83 \( 1 + (76.6 - 76.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 41.0T + 7.92e3T^{2} \)
97 \( 1 + (-81.7 - 81.7i)T + 9.40e3iT^{2} \)
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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

−15.67320462962645132711413332474, −14.33893181672884078930430397319, −12.54628622245091081736491169587, −11.77537047291586914383487780997, −10.75177445247353756773601961622, −9.521719546007788997280243555052, −7.85363033204737597158222444516, −6.99739291332937950548884940310, −5.77920211189023878915210953869, −2.61448766566261367109554987358, 0.26707032984705373146136416608, 3.76194666675450833951294183831, 5.66790470042630145697798211277, 7.32046481301254405695919034786, 8.621330730192359403058655505375, 10.02019735884697809625871082951, 10.58402221868743756019890905484, 12.02827918332618988051620909850, 12.81200617304594223604708809343, 15.04247436292919210421021324354

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