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Properties

Label 2-60-20.19-c6-0-35
Degree $2$
Conductor $60$
Sign $-0.971 + 0.237i$
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 + (−818. − 575. 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 + (−0.818 − 0.575i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.971 + 0.237i$
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: \((2,\ 60,\ (\ :3),\ -0.971 + 0.237i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.255451 - 2.12351i\)
\(L(\frac12)\) \(\approx\) \(0.255451 - 2.12351i\)
\(L(4)\) 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

−12.91909613282265232666967624513, −12.71987859958701699999475235054, −11.20116686165022074234992623720, −9.619086718839056163311464564120, −9.111259751544166039046207914119, −7.10228570272009588290165264241, −5.35480835664880113299131825093, −4.01341442047810856460392947096, −2.53943754149657026526735636030, −0.65960679239314621331570483888, 2.88152066036772575808672988238, 3.82156220356915080494677846834, 5.96529504877398732246659344516, 6.92262846883461170821553456405, 8.157340704936187556644578927239, 9.537431376996081630632132544909, 11.01298781339459568035917458278, 12.54143091469536679097825179926, 13.45632376174536258719955015821, 14.45103996292045807260625130488

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