↑

Properties

Label 2-60-20.19-c6-0-1
Degree $2$
Conductor $60$
Sign $-0.817 - 0.576i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.93 + 1.03i)2-s + 15.5·3-s + (61.8 − 16.4i)4-s + (−80.2 − 95.8i)5-s + (−123. + 16.1i)6-s − 143.·7-s + (−473. + 194. i)8-s + 243·9-s + (735. + 677. i)10-s + 1.56e3i·11-s + (964. − 256. i)12-s − 3.87e3i·13-s + (1.13e3 − 148. i)14-s + (−1.25e3 − 1.49e3i)15-s + (3.55e3 − 2.03e3i)16-s + 5.03e3i·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.129i)2-s + 0.577·3-s + (0.966 − 0.256i)4-s + (−0.641 − 0.766i)5-s + (−0.572 + 0.0747i)6-s − 0.418·7-s + (−0.925 + 0.379i)8-s + 0.333·9-s + (0.735 + 0.677i)10-s + 1.17i·11-s + (0.558 − 0.148i)12-s − 1.76i·13-s + (0.415 − 0.0541i)14-s + (−0.370 − 0.442i)15-s + (0.868 − 0.496i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.817 - 0.576i$
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: \((2,\ 60,\ (\ :3),\ -0.817 - 0.576i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0799792 + 0.252170i\)
\(L(\frac12)\) \(\approx\) \(0.0799792 + 0.252170i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.93 - 1.03i)T \)
3 \( 1 - 15.5T \)
5 \( 1 + (80.2 + 95.8i)T \)
good7 \( 1 + 143.T + 1.17e5T^{2} \)
11 \( 1 - 1.56e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.87e3iT - 4.82e6T^{2} \)
17 \( 1 - 5.03e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.17e4iT - 4.70e7T^{2} \)
23 \( 1 + 2.17e4T + 1.48e8T^{2} \)
29 \( 1 + 4.01e4T + 5.94e8T^{2} \)
31 \( 1 + 223. iT - 8.87e8T^{2} \)
37 \( 1 - 1.73e4iT - 2.56e9T^{2} \)
41 \( 1 - 2.19e3T + 4.75e9T^{2} \)
43 \( 1 + 3.19e3T + 6.32e9T^{2} \)
47 \( 1 - 1.04e4T + 1.07e10T^{2} \)
53 \( 1 - 1.99e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.49e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.26e5T + 5.15e10T^{2} \)
67 \( 1 + 2.35e5T + 9.04e10T^{2} \)
71 \( 1 + 1.70e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.45e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.73e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.80e5T + 3.26e11T^{2} \)
89 \( 1 - 4.21e5T + 4.96e11T^{2} \)
97 \( 1 + 8.55e5iT - 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.77868800957098628208746116531, −12.81838124582238146632248426424, −12.18672263204136346190838366529, −10.42134789954024908818037479001, −9.618847930460091046503721106337, −8.162331646163256281093059784045, −7.69868876004929786050991785432, −5.81295390794051569727291932385, −3.68467503405804697181491459337, −1.70754068766639716586737522621, 0.13367944139075584562874053337, 2.37460715873329253710466210170, 3.71016866367055243893379498999, 6.50172699106286118051635341393, 7.43578975016207833230111514299, 8.760266459822230780372467190566, 9.661576534621176524526602196600, 11.15738746583616829631573588535, 11.73411709596062837508443216749, 13.54553284877022771157582798995

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