Properties

Label 2-60-4.3-c6-0-9
Degree $2$
Conductor $60$
Sign $0.793 + 0.608i$
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  + (−3.53 − 7.17i)2-s − 15.5i·3-s + (−38.9 + 50.7i)4-s + 55.9·5-s + (−111. + 55.1i)6-s + 99.9i·7-s + (502. + 99.7i)8-s − 243·9-s + (−197. − 401. i)10-s + 2.39e3i·11-s + (791. + 607. i)12-s + 3.61e3·13-s + (717. − 353. i)14-s − 871. i·15-s + (−1.06e3 − 3.95e3i)16-s − 927.·17-s + ⋯
L(s)  = 1  + (−0.442 − 0.896i)2-s − 0.577i·3-s + (−0.608 + 0.793i)4-s + 0.447·5-s + (−0.517 + 0.255i)6-s + 0.291i·7-s + (0.980 + 0.194i)8-s − 0.333·9-s + (−0.197 − 0.401i)10-s + 1.80i·11-s + (0.458 + 0.351i)12-s + 1.64·13-s + (0.261 − 0.128i)14-s − 0.258i·15-s + (−0.259 − 0.965i)16-s − 0.188·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.40953 - 0.478312i\)
\(L(\frac12)\) \(\approx\) \(1.40953 - 0.478312i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.53 + 7.17i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 - 99.9iT - 1.17e5T^{2} \)
11 \( 1 - 2.39e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.61e3T + 4.82e6T^{2} \)
17 \( 1 + 927.T + 2.41e7T^{2} \)
19 \( 1 + 773. iT - 4.70e7T^{2} \)
23 \( 1 + 1.22e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.90e4T + 5.94e8T^{2} \)
31 \( 1 + 9.56e3iT - 8.87e8T^{2} \)
37 \( 1 - 8.86e4T + 2.56e9T^{2} \)
41 \( 1 + 5.37e4T + 4.75e9T^{2} \)
43 \( 1 - 1.40e5iT - 6.32e9T^{2} \)
47 \( 1 - 1.05e5iT - 1.07e10T^{2} \)
53 \( 1 + 8.09e4T + 2.21e10T^{2} \)
59 \( 1 + 5.90e4iT - 4.21e10T^{2} \)
61 \( 1 - 1.16e5T + 5.15e10T^{2} \)
67 \( 1 - 1.34e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.68e5iT - 1.28e11T^{2} \)
73 \( 1 - 7.95e4T + 1.51e11T^{2} \)
79 \( 1 + 6.70e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.49e5iT - 3.26e11T^{2} \)
89 \( 1 - 7.85e5T + 4.96e11T^{2} \)
97 \( 1 - 6.64e5T + 8.32e11T^{2} \)
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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

−13.28444681603798832321154157055, −12.61843021097311262777909638470, −11.48600687803255552982369991326, −10.28059569128370841606834588008, −9.145837925446222793467756726196, −7.976033729226102232339121210852, −6.46488962589208908367642907082, −4.48252344434987165575392480181, −2.53819453598751962433351990214, −1.25070369789294193795967737242, 0.898274124129207144325038224100, 3.70180278941629151846040041131, 5.50103843303988356230073503322, 6.40905036640162290630849683916, 8.243604684570078109084818248994, 9.014461638748422407711912406293, 10.38074418031601841900417904951, 11.22947320557509227096056811910, 13.58300436754372106352529381773, 13.83608228375675984496859273621

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