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Properties

Label 2-60-12.11-c3-0-18
Degree $2$
Conductor $60$
Sign $0.886 + 0.461i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 + 0.245i)2-s + (1.56 − 4.95i)3-s + (7.87 + 1.38i)4-s + 5i·5-s + (5.63 − 13.5i)6-s − 5.80i·7-s + (21.8 + 5.83i)8-s + (−22.0 − 15.5i)9-s + (−1.22 + 14.0i)10-s + 27.6·11-s + (19.1 − 36.8i)12-s − 70.9·13-s + (1.42 − 16.3i)14-s + (24.7 + 7.83i)15-s + (60.1 + 21.8i)16-s + 89.8i·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0868i)2-s + (0.301 − 0.953i)3-s + (0.984 + 0.172i)4-s + 0.447i·5-s + (0.383 − 0.923i)6-s − 0.313i·7-s + (0.966 + 0.257i)8-s + (−0.818 − 0.574i)9-s + (−0.0388 + 0.445i)10-s + 0.757·11-s + (0.461 − 0.886i)12-s − 1.51·13-s + (0.0272 − 0.312i)14-s + (0.426 + 0.134i)15-s + (0.940 + 0.340i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.886 + 0.461i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.886 + 0.461i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.53396 - 0.620256i\)
\(L(\frac12)\) \(\approx\) \(2.53396 - 0.620256i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 - 0.245i)T \)
3 \( 1 + (-1.56 + 4.95i)T \)
5 \( 1 - 5iT \)
good7 \( 1 + 5.80iT - 343T^{2} \)
11 \( 1 - 27.6T + 1.33e3T^{2} \)
13 \( 1 + 70.9T + 2.19e3T^{2} \)
17 \( 1 - 89.8iT - 4.91e3T^{2} \)
19 \( 1 - 68.6iT - 6.85e3T^{2} \)
23 \( 1 + 73.5T + 1.21e4T^{2} \)
29 \( 1 + 18.2iT - 2.43e4T^{2} \)
31 \( 1 + 277. iT - 2.97e4T^{2} \)
37 \( 1 + 26.7T + 5.06e4T^{2} \)
41 \( 1 - 364. iT - 6.89e4T^{2} \)
43 \( 1 + 425. iT - 7.95e4T^{2} \)
47 \( 1 - 80.9T + 1.03e5T^{2} \)
53 \( 1 + 610. iT - 1.48e5T^{2} \)
59 \( 1 - 522.T + 2.05e5T^{2} \)
61 \( 1 - 372.T + 2.26e5T^{2} \)
67 \( 1 + 369. iT - 3.00e5T^{2} \)
71 \( 1 + 985.T + 3.57e5T^{2} \)
73 \( 1 + 2.29T + 3.89e5T^{2} \)
79 \( 1 + 175. iT - 4.93e5T^{2} \)
83 \( 1 + 207.T + 5.71e5T^{2} \)
89 \( 1 - 186. iT - 7.04e5T^{2} \)
97 \( 1 + 1.24e3T + 9.12e5T^{2} \)
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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

−14.53571057183108727806721034446, −13.43264207363759403216891221069, −12.38872647120275109952018615652, −11.59969883638107918180522451335, −10.06525165107973714915651757199, −8.034639588979335830482217022890, −7.00836091591424804440832728264, −5.90946455358657838384341807101, −3.85800664834443107691289209760, −2.11750880999031894296508581268, 2.71506110281388789899373757605, 4.41545887606059743285653396852, 5.37430807931593089085742037795, 7.20860254469942431904760560956, 9.001435076223832552680984972815, 10.10482935525143994934799508182, 11.51966697671628281383619916990, 12.34969347627131045210212625962, 13.85775556755989495932711442275, 14.56480238147332242497330247557

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