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Properties

Label 2-60-5.3-c6-0-0
Degree $2$
Conductor $60$
Sign $0.363 - 0.931i$
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  + (−11.0 − 11.0i)3-s + (−102. − 71.0i)5-s + (−79.3 + 79.3i)7-s + 242. i·9-s + 212.·11-s + (1.38e3 + 1.38e3i)13-s + (351. + 1.91e3i)15-s + (312. − 312. i)17-s + 7.90e3i·19-s + 1.75e3·21-s + (8.39e3 + 8.39e3i)23-s + (5.54e3 + 1.46e4i)25-s + (2.67e3 − 2.67e3i)27-s + 2.55e4i·29-s − 6.83e3·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.822 − 0.568i)5-s + (−0.231 + 0.231i)7-s + 0.333i·9-s + 0.159·11-s + (0.629 + 0.629i)13-s + (0.104 + 0.567i)15-s + (0.0635 − 0.0635i)17-s + 1.15i·19-s + 0.188·21-s + (0.690 + 0.690i)23-s + (0.354 + 0.935i)25-s + (0.136 − 0.136i)27-s + 1.04i·29-s − 0.229·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.363 - 0.931i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.731290 + 0.499476i\)
\(L(\frac12)\) \(\approx\) \(0.731290 + 0.499476i\)
\(L(4)\) 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 \( 1 + (11.0 + 11.0i)T \)
5 \( 1 + (102. + 71.0i)T \)
good7 \( 1 + (79.3 - 79.3i)T - 1.17e5iT^{2} \)
11 \( 1 - 212.T + 1.77e6T^{2} \)
13 \( 1 + (-1.38e3 - 1.38e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-312. + 312. i)T - 2.41e7iT^{2} \)
19 \( 1 - 7.90e3iT - 4.70e7T^{2} \)
23 \( 1 + (-8.39e3 - 8.39e3i)T + 1.48e8iT^{2} \)
29 \( 1 - 2.55e4iT - 5.94e8T^{2} \)
31 \( 1 + 6.83e3T + 8.87e8T^{2} \)
37 \( 1 + (1.89e4 - 1.89e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 4.20e4T + 4.75e9T^{2} \)
43 \( 1 + (-1.15e3 - 1.15e3i)T + 6.32e9iT^{2} \)
47 \( 1 + (-1.18e5 + 1.18e5i)T - 1.07e10iT^{2} \)
53 \( 1 + (1.29e5 + 1.29e5i)T + 2.21e10iT^{2} \)
59 \( 1 + 2.14e5iT - 4.21e10T^{2} \)
61 \( 1 + 4.15e5T + 5.15e10T^{2} \)
67 \( 1 + (1.10e5 - 1.10e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 1.29e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.41e5 - 3.41e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 7.47e5iT - 2.43e11T^{2} \)
83 \( 1 + (-5.75e5 - 5.75e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 7.93e5iT - 4.96e11T^{2} \)
97 \( 1 + (7.72e5 - 7.72e5i)T - 8.32e11iT^{2} \)
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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

−13.93538554723066749404891494250, −12.70779451931886285839776779807, −11.92653147451654663400923879655, −10.89143784690015099545984283888, −9.227122750871200324616957192133, −8.064431173988940206892707144412, −6.75842092322843544363874805212, −5.27354320958810862776359850434, −3.67877728541937454634892766185, −1.34196028728867239601102578114, 0.43053541309147419251628204276, 3.13500818440602260651323989794, 4.50021836193463136850025307454, 6.22833583503177851949452677643, 7.48611872975277565603693959682, 8.935666424797008791660269689052, 10.45711843742778227537911852203, 11.18194187651885570721657826696, 12.34809534651521427159662685565, 13.65037978577586043237474000514

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