Properties

Label 2-60-20.19-c6-0-28
Degree $2$
Conductor $60$
Sign $-0.901 + 0.431i$
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 + (35.0 − 999. 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 + (0.0350 − 0.999i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.901 + 0.431i$
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: \((2,\ 60,\ (\ :3),\ -0.901 + 0.431i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.219757 - 0.967518i\)
\(L(\frac12)\) \(\approx\) \(0.219757 - 0.967518i\)
\(L(4)\) 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

−13.34208018495080254323970207576, −12.39575058346857640017681342602, −10.69529466835355961111352050503, −9.836954523018978452963455563134, −9.095604920021937446338313118845, −7.49341607980856661203002635080, −5.89996678127426681033935096971, −3.37164098979780696342260359407, −2.67313579165604154683243461468, −0.43076385221055329937648443237, 1.83831352351837450226697748475, 4.18791127938713178838309307541, 6.08153762306009162230478616328, 6.88415225026001488314824277756, 8.602741429131448292546831667531, 9.611412216248188474007351400514, 10.06008749656047382355382473614, 12.57481314751059426457957959402, 13.38353834369989014373775078174, 14.36615004388143160621553150636

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