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Properties

Label 2-60-60.47-c2-0-8
Degree $2$
Conductor $60$
Sign $0.289 - 0.957i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 + 1.07i)2-s + (−0.130 + 2.99i)3-s + (1.67 + 3.63i)4-s + (−1.65 − 4.71i)5-s + (−3.45 + 4.90i)6-s + (1.91 − 1.91i)7-s + (−1.11 + 7.92i)8-s + (−8.96 − 0.782i)9-s + (2.31 − 9.72i)10-s + 6.87·11-s + (−11.1 + 4.53i)12-s + (12.2 − 12.2i)13-s + (5.29 − 1.15i)14-s + (14.3 − 4.33i)15-s + (−10.4 + 12.1i)16-s + (9.47 − 9.47i)17-s + ⋯
L(s)  = 1  + (0.841 + 0.539i)2-s + (−0.0434 + 0.999i)3-s + (0.417 + 0.908i)4-s + (−0.330 − 0.943i)5-s + (−0.575 + 0.817i)6-s + (0.273 − 0.273i)7-s + (−0.138 + 0.990i)8-s + (−0.996 − 0.0869i)9-s + (0.231 − 0.972i)10-s + 0.624·11-s + (−0.925 + 0.377i)12-s + (0.944 − 0.944i)13-s + (0.378 − 0.0826i)14-s + (0.957 − 0.288i)15-s + (−0.651 + 0.758i)16-s + (0.557 − 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37902 + 1.02383i\)
\(L(\frac12)\) \(\approx\) \(1.37902 + 1.02383i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.68 - 1.07i)T \)
3 \( 1 + (0.130 - 2.99i)T \)
5 \( 1 + (1.65 + 4.71i)T \)
good7 \( 1 + (-1.91 + 1.91i)T - 49iT^{2} \)
11 \( 1 - 6.87T + 121T^{2} \)
13 \( 1 + (-12.2 + 12.2i)T - 169iT^{2} \)
17 \( 1 + (-9.47 + 9.47i)T - 289iT^{2} \)
19 \( 1 + 33.2T + 361T^{2} \)
23 \( 1 + (7.20 - 7.20i)T - 529iT^{2} \)
29 \( 1 - 2.29T + 841T^{2} \)
31 \( 1 - 12.1iT - 961T^{2} \)
37 \( 1 + (20.7 + 20.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 + (15.1 + 15.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-26.7 - 26.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-15.5 - 15.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 63.0iT - 3.48e3T^{2} \)
61 \( 1 - 28.4T + 3.72e3T^{2} \)
67 \( 1 + (32.4 - 32.4i)T - 4.48e3iT^{2} \)
71 \( 1 + 88.8T + 5.04e3T^{2} \)
73 \( 1 + (-71.1 + 71.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 75.1T + 6.24e3T^{2} \)
83 \( 1 + (-58.6 + 58.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 41.1T + 7.92e3T^{2} \)
97 \( 1 + (-30.3 - 30.3i)T + 9.40e3iT^{2} \)
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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

−15.18205566097580414359197824381, −14.14533117249101755863411711145, −12.95048213696038916472778008260, −11.82385665618003986192255906813, −10.70500755454900040538847090405, −8.948351715213818848111141027358, −8.004133566125378102373897630914, −6.02034487217630432333277299324, −4.76072477105531461636585954037, −3.67053098191343481472846751205, 2.03512803224587814105581809619, 3.85816482106648668424191426706, 6.06319544965495805130069333949, 6.85689519605348898444702854729, 8.578228383927319341653234144454, 10.54213631747025712510086610041, 11.49768269635719340892827690538, 12.28154849269766631291536188775, 13.53936076037602200323410538848, 14.40803133352736431265830779790

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