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Properties

Label 2-60-60.23-c2-0-15
Degree $2$
Conductor $60$
Sign $-0.0776 + 0.996i$
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  + (−0.770 + 1.84i)2-s + (−1.12 − 2.78i)3-s + (−2.81 − 2.84i)4-s + (−3.86 − 3.17i)5-s + (5.99 + 0.0674i)6-s + (−4.75 − 4.75i)7-s + (7.41 − 2.99i)8-s + (−6.46 + 6.25i)9-s + (8.83 − 4.68i)10-s + 11.9·11-s + (−4.74 + 11.0i)12-s + (−4.22 − 4.22i)13-s + (12.4 − 5.10i)14-s + (−4.48 + 14.3i)15-s + (−0.188 + 15.9i)16-s + (−9.35 − 9.35i)17-s + ⋯
L(s)  = 1  + (−0.385 + 0.922i)2-s + (−0.375 − 0.927i)3-s + (−0.702 − 0.711i)4-s + (−0.772 − 0.635i)5-s + (0.999 + 0.0112i)6-s + (−0.678 − 0.678i)7-s + (0.927 − 0.374i)8-s + (−0.718 + 0.695i)9-s + (0.883 − 0.468i)10-s + 1.08·11-s + (−0.395 + 0.918i)12-s + (−0.324 − 0.324i)13-s + (0.887 − 0.364i)14-s + (−0.299 + 0.954i)15-s + (−0.0117 + 0.999i)16-s + (−0.550 − 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0776 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0776 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0776 + 0.996i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.0776 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.354376 - 0.383061i\)
\(L(\frac12)\) \(\approx\) \(0.354376 - 0.383061i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.770 - 1.84i)T \)
3 \( 1 + (1.12 + 2.78i)T \)
5 \( 1 + (3.86 + 3.17i)T \)
good7 \( 1 + (4.75 + 4.75i)T + 49iT^{2} \)
11 \( 1 - 11.9T + 121T^{2} \)
13 \( 1 + (4.22 + 4.22i)T + 169iT^{2} \)
17 \( 1 + (9.35 + 9.35i)T + 289iT^{2} \)
19 \( 1 + 1.48T + 361T^{2} \)
23 \( 1 + (11.6 + 11.6i)T + 529iT^{2} \)
29 \( 1 - 39.3T + 841T^{2} \)
31 \( 1 + 43.6iT - 961T^{2} \)
37 \( 1 + (-49.1 + 49.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 27.0iT - 1.68e3T^{2} \)
43 \( 1 + (8.84 - 8.84i)T - 1.84e3iT^{2} \)
47 \( 1 + (15.0 - 15.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (14.6 - 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 61.7iT - 3.48e3T^{2} \)
61 \( 1 + 84.6T + 3.72e3T^{2} \)
67 \( 1 + (-65.7 - 65.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 14.2T + 5.04e3T^{2} \)
73 \( 1 + (-16.0 - 16.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 9.32T + 6.24e3T^{2} \)
83 \( 1 + (12.7 + 12.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.4T + 7.92e3T^{2} \)
97 \( 1 + (-6.90 + 6.90i)T - 9.40e3iT^{2} \)
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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

−14.57454347065246599494192672803, −13.46865071304590672876912742816, −12.51644878843718210495896967858, −11.22447480572418045917861904324, −9.585228965852253334860272852199, −8.269482681741931309449301547104, −7.22665143965516506717955602192, −6.20546293989677576822796084479, −4.43553624422490169569061959054, −0.59903320505925603494612498018, 3.15564526505518902145396371766, 4.39471462521306530321055053976, 6.52445478845797795708509440337, 8.510196758926229387521920989183, 9.547603909878553932971448179698, 10.59487828979137085896522608079, 11.69958015684721189528169511352, 12.27757908028898527403942635161, 14.09784910272911977579614635027, 15.24983844471596350465909525918

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