Properties

Label 2-60-4.3-c6-0-1
Degree $2$
Conductor $60$
Sign $0.0829 + 0.996i$
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  + (0.331 + 7.99i)2-s + 15.5i·3-s + (−63.7 + 5.30i)4-s − 55.9·5-s + (−124. + 5.17i)6-s + 552. i·7-s + (−63.5 − 508. i)8-s − 243·9-s + (−18.5 − 446. i)10-s − 1.28e3i·11-s + (−82.7 − 994. i)12-s − 2.51e3·13-s + (−4.41e3 + 183. i)14-s − 871. i·15-s + (4.03e3 − 676. i)16-s + 4.59e3·17-s + ⋯
L(s)  = 1  + (0.0414 + 0.999i)2-s + 0.577i·3-s + (−0.996 + 0.0829i)4-s − 0.447·5-s + (−0.576 + 0.0239i)6-s + 1.60i·7-s + (−0.124 − 0.992i)8-s − 0.333·9-s + (−0.0185 − 0.446i)10-s − 0.965i·11-s + (−0.0478 − 0.575i)12-s − 1.14·13-s + (−1.60 + 0.0667i)14-s − 0.258i·15-s + (0.986 − 0.165i)16-s + 0.935·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0829 + 0.996i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.0829 + 0.996i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.202175 - 0.186050i\)
\(L(\frac12)\) \(\approx\) \(0.202175 - 0.186050i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.331 - 7.99i)T \)
3 \( 1 - 15.5iT \)
5 \( 1 + 55.9T \)
good7 \( 1 - 552. iT - 1.17e5T^{2} \)
11 \( 1 + 1.28e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.51e3T + 4.82e6T^{2} \)
17 \( 1 - 4.59e3T + 2.41e7T^{2} \)
19 \( 1 + 9.80e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.52e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.43e4T + 5.94e8T^{2} \)
31 \( 1 + 3.44e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.14e4T + 2.56e9T^{2} \)
41 \( 1 + 3.76e4T + 4.75e9T^{2} \)
43 \( 1 + 1.53e3iT - 6.32e9T^{2} \)
47 \( 1 - 1.31e5iT - 1.07e10T^{2} \)
53 \( 1 + 2.28e5T + 2.21e10T^{2} \)
59 \( 1 - 2.71e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.52e5T + 5.15e10T^{2} \)
67 \( 1 + 7.17e4iT - 9.04e10T^{2} \)
71 \( 1 + 1.93e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.25e5T + 1.51e11T^{2} \)
79 \( 1 - 6.09e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.92e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.54e5T + 4.96e11T^{2} \)
97 \( 1 + 1.46e6T + 8.32e11T^{2} \)
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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

−15.14759099950926607933131547075, −13.92586504467722143640630154206, −12.50079640466554993007777448230, −11.43552407097029540403575338866, −9.599638919967430150913153959019, −8.819914662151858183981743986263, −7.59818007795872532832633449267, −5.87929315886679713054378730639, −4.97051804345147032861270544935, −3.14633926545379518070777509663, 0.11412494824800675948380878915, 1.61493267711635289180160181245, 3.53420984483801053073431435718, 4.84387452155881615377696602247, 7.09524194516886329495255272518, 8.079823345184872981663215679080, 9.906405141601502351058193671616, 10.59533640065395326789000629763, 12.12387861241103739028595849423, 12.63053118285361916995357548196

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