Properties

Label 2-60-20.19-c6-0-32
Degree $2$
Conductor $60$
Sign $0.423 + 0.905i$
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  + (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 + (−537. − 843. 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 + (−0.537 − 0.843i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.423 + 0.905i$
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: \((2,\ 60,\ (\ :3),\ 0.423 + 0.905i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.10658 - 1.34037i\)
\(L(\frac12)\) \(\approx\) \(2.10658 - 1.34037i\)
\(L(4)\) 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 \( 1 + 8.55e5iT - 8.32e11T^{2} \)
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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

−13.20396427536702579837250439561, −12.83272096380140334795247230707, −11.37094523857519403762440099206, −10.86112638485477079407389799843, −8.601273241528997789939775618151, −7.42266724544859449885999479016, −5.75753143806467664992751046131, −4.85763275711633553474914171643, −3.29777815085289673043143156971, −0.846424273683988079130461932608, 1.90819921033868639131863578011, 3.84365961505398471841663973911, 5.00651458498351527118003544932, 6.69228883526938767069982446847, 7.43416859771557335029738705878, 9.762773544329373608861165984625, 11.16736297607929746462145497033, 11.68892849420199125971900688829, 12.76120502050840586250496260432, 14.29409823302609955041700946867

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