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Properties

Label 2-60-20.19-c6-0-23
Degree $2$
Conductor $60$
Sign $0.0905 + 0.995i$
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  + (3.70 − 7.08i)2-s − 15.5·3-s + (−36.5 − 52.5i)4-s + (108. + 61.7i)5-s + (−57.7 + 110. i)6-s + 629.·7-s + (−507. + 64.1i)8-s + 243·9-s + (840. − 541. i)10-s + 1.25e3i·11-s + (569. + 819. i)12-s − 3.33e3i·13-s + (2.33e3 − 4.45e3i)14-s + (−1.69e3 − 962. i)15-s + (−1.42e3 + 3.83e3i)16-s − 7.63e3i·17-s + ⋯
L(s)  = 1  + (0.463 − 0.886i)2-s − 0.577·3-s + (−0.570 − 0.821i)4-s + (0.869 + 0.494i)5-s + (−0.267 + 0.511i)6-s + 1.83·7-s + (−0.992 + 0.125i)8-s + 0.333·9-s + (0.840 − 0.541i)10-s + 0.943i·11-s + (0.329 + 0.474i)12-s − 1.51i·13-s + (0.849 − 1.62i)14-s + (−0.501 − 0.285i)15-s + (−0.348 + 0.937i)16-s − 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0905 + 0.995i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0905 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0905 + 0.995i$
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: \((2,\ 60,\ (\ :3),\ 0.0905 + 0.995i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.78941 - 1.63415i\)
\(L(\frac12)\) \(\approx\) \(1.78941 - 1.63415i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.70 + 7.08i)T \)
3 \( 1 + 15.5T \)
5 \( 1 + (-108. - 61.7i)T \)
good7 \( 1 - 629.T + 1.17e5T^{2} \)
11 \( 1 - 1.25e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.33e3iT - 4.82e6T^{2} \)
17 \( 1 + 7.63e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.25e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.44e4T + 1.48e8T^{2} \)
29 \( 1 + 7.20e3T + 5.94e8T^{2} \)
31 \( 1 - 3.24e3iT - 8.87e8T^{2} \)
37 \( 1 + 5.48e4iT - 2.56e9T^{2} \)
41 \( 1 - 2.91e4T + 4.75e9T^{2} \)
43 \( 1 - 1.27e4T + 6.32e9T^{2} \)
47 \( 1 + 1.50e4T + 1.07e10T^{2} \)
53 \( 1 - 2.03e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.45e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.11e5T + 5.15e10T^{2} \)
67 \( 1 + 1.01e5T + 9.04e10T^{2} \)
71 \( 1 - 5.30e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.24e5iT - 1.51e11T^{2} \)
79 \( 1 - 1.37e4iT - 2.43e11T^{2} \)
83 \( 1 - 6.32e5T + 3.26e11T^{2} \)
89 \( 1 + 3.03e5T + 4.96e11T^{2} \)
97 \( 1 - 6.25e5iT - 8.32e11T^{2} \)
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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

−13.50624977257431213690037709335, −12.33784734084535237514535814968, −11.16404609519902520660276712041, −10.55818603221138384006990777046, −9.281329408703183409055093447117, −7.40528291300686015385931202812, −5.50200021407494669088495900753, −4.80246190940215780769966623095, −2.54550729397927237273272183175, −1.13244107902215792155093828040, 1.53525149682804163182191377495, 4.36793835338556740682687578212, 5.35976422380104873422529302911, 6.46955214853477247976348327649, 8.114647554221361290229077230384, 9.035482138772470898067445984147, 10.91554805359738332560056162396, 11.95054152329593359683459976960, 13.27276148570903957307611818493, 14.18320600017053612122880583979

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