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Properties

Label 2-60-20.19-c6-0-27
Degree $2$
Conductor $60$
Sign $-0.999 - 0.0272i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.26 − 6.02i)2-s − 15.5·3-s + (−8.50 + 63.4i)4-s + (−13.2 − 124. i)5-s + (82.1 + 93.8i)6-s + 380.·7-s + (426. − 282. i)8-s + 243·9-s + (−678. + 734. i)10-s − 847. i·11-s + (132. − 988. i)12-s − 311. i·13-s + (−2.00e3 − 2.28e3i)14-s + (206. + 1.93e3i)15-s + (−3.95e3 − 1.07e3i)16-s − 4.50e3i·17-s + ⋯
L(s)  = 1  + (−0.658 − 0.752i)2-s − 0.577·3-s + (−0.132 + 0.991i)4-s + (−0.105 − 0.994i)5-s + (0.380 + 0.434i)6-s + 1.10·7-s + (0.833 − 0.552i)8-s + 0.333·9-s + (−0.678 + 0.734i)10-s − 0.636i·11-s + (0.0767 − 0.572i)12-s − 0.141i·13-s + (−0.729 − 0.833i)14-s + (0.0611 + 0.574i)15-s + (−0.964 − 0.263i)16-s − 0.917i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0272i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.0272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.999 - 0.0272i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.999 - 0.0272i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00907762 + 0.666588i\)
\(L(\frac12)\) \(\approx\) \(0.00907762 + 0.666588i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.26 + 6.02i)T \)
3 \( 1 + 15.5T \)
5 \( 1 + (13.2 + 124. i)T \)
good7 \( 1 - 380.T + 1.17e5T^{2} \)
11 \( 1 + 847. iT - 1.77e6T^{2} \)
13 \( 1 + 311. iT - 4.82e6T^{2} \)
17 \( 1 + 4.50e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.56e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.54e4T + 1.48e8T^{2} \)
29 \( 1 + 1.88e4T + 5.94e8T^{2} \)
31 \( 1 + 460. iT - 8.87e8T^{2} \)
37 \( 1 + 9.85e4iT - 2.56e9T^{2} \)
41 \( 1 - 6.96e4T + 4.75e9T^{2} \)
43 \( 1 + 1.45e5T + 6.32e9T^{2} \)
47 \( 1 - 3.36e4T + 1.07e10T^{2} \)
53 \( 1 + 1.28e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.14e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.54e5T + 5.15e10T^{2} \)
67 \( 1 + 5.23e5T + 9.04e10T^{2} \)
71 \( 1 - 1.65e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.08e5iT - 1.51e11T^{2} \)
79 \( 1 + 1.01e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.88e4T + 3.26e11T^{2} \)
89 \( 1 + 1.05e6T + 4.96e11T^{2} \)
97 \( 1 + 6.31e5iT - 8.32e11T^{2} \)
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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

−12.89654945166277671627328811552, −11.85813048230057979897864563994, −11.18372310836629409943657578532, −9.838847325818107428401403120191, −8.572681615930970149394418114681, −7.64184088144277668477060763381, −5.44280008803205813870029649082, −4.07402530114857012039786777375, −1.73662217729562546590774918271, −0.37719876806980152817708690844, 1.76517937821377487237987467999, 4.55628066015959400276323520084, 6.03395488727609623784975859941, 7.18621047066958332748819570804, 8.225955628989699397988900258021, 9.891445084714548926508495958480, 10.83279375461165995827935373321, 11.75359070940835585874643466039, 13.64796102021644080505768150263, 14.82621947664930213532155676902

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