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Properties

Label 2-60-15.14-c6-0-0
Degree $2$
Conductor $60$
Sign $-0.987 - 0.160i$
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  + (−22.8 + 14.3i)3-s + (93.9 + 82.4i)5-s + 279. i·7-s + (318. − 655. i)9-s − 772. i·11-s + 2.87e3i·13-s + (−3.33e3 − 541. i)15-s − 2.79e3·17-s − 1.30e4·19-s + (−4.00e3 − 6.40e3i)21-s − 1.05e4·23-s + (2.02e3 + 1.54e4i)25-s + (2.10e3 + 1.95e4i)27-s − 3.72e4i·29-s − 8.71e3·31-s + ⋯
L(s)  = 1  + (−0.847 + 0.530i)3-s + (0.751 + 0.659i)5-s + 0.815i·7-s + (0.436 − 0.899i)9-s − 0.580i·11-s + 1.30i·13-s + (−0.987 − 0.160i)15-s − 0.568·17-s − 1.90·19-s + (−0.432 − 0.691i)21-s − 0.867·23-s + (0.129 + 0.991i)25-s + (0.107 + 0.994i)27-s − 1.52i·29-s − 0.292·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.987 - 0.160i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.987 - 0.160i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0623283 + 0.772259i\)
\(L(\frac12)\) \(\approx\) \(0.0623283 + 0.772259i\)
\(L(4)\) 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 \( 1 + (22.8 - 14.3i)T \)
5 \( 1 + (-93.9 - 82.4i)T \)
good7 \( 1 - 279. iT - 1.17e5T^{2} \)
11 \( 1 + 772. iT - 1.77e6T^{2} \)
13 \( 1 - 2.87e3iT - 4.82e6T^{2} \)
17 \( 1 + 2.79e3T + 2.41e7T^{2} \)
19 \( 1 + 1.30e4T + 4.70e7T^{2} \)
23 \( 1 + 1.05e4T + 1.48e8T^{2} \)
29 \( 1 + 3.72e4iT - 5.94e8T^{2} \)
31 \( 1 + 8.71e3T + 8.87e8T^{2} \)
37 \( 1 - 3.55e3iT - 2.56e9T^{2} \)
41 \( 1 - 1.08e5iT - 4.75e9T^{2} \)
43 \( 1 + 6.66e4iT - 6.32e9T^{2} \)
47 \( 1 - 4.02e3T + 1.07e10T^{2} \)
53 \( 1 + 7.95e4T + 2.21e10T^{2} \)
59 \( 1 - 1.51e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.07e5T + 5.15e10T^{2} \)
67 \( 1 - 3.63e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.79e4iT - 1.28e11T^{2} \)
73 \( 1 + 4.61e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.27e5T + 2.43e11T^{2} \)
83 \( 1 + 9.98e5T + 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 - 7.34e5iT - 8.32e11T^{2} \)
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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

−14.53432740086505556526521234137, −13.24163263103963158159236599253, −11.85782515652739016374776945285, −10.99749151468025863841688255296, −9.869980495230083300306982723439, −8.779424671899337748333474502400, −6.57597048390701085655156344617, −5.89381690909960291840023637689, −4.23852871109792212087018980216, −2.17851064239067404130331483130, 0.33881634266145155064583879628, 1.86720050819828320515858177328, 4.50863091995492953381207667709, 5.78349925400115093962910066435, 7.01637819724301595244630270801, 8.422391732675055052855885932489, 10.13191880135056182412428458229, 10.85709868840444097861063395409, 12.61450849706030613973762175607, 12.91730357635843950958368133320

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