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Properties

Label 2-60-60.47-c2-0-17
Degree $2$
Conductor $60$
Sign $-0.0687 + 0.997i$
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  + (0.837 − 1.81i)2-s + (1.32 − 2.69i)3-s + (−2.59 − 3.04i)4-s + (3.21 + 3.82i)5-s + (−3.78 − 4.65i)6-s + (−3.54 + 3.54i)7-s + (−7.70 + 2.16i)8-s + (−5.50 − 7.12i)9-s + (9.64 − 2.63i)10-s + 16.8·11-s + (−11.6 + 2.96i)12-s + (−8.64 + 8.64i)13-s + (3.46 + 9.40i)14-s + (14.5 − 3.59i)15-s + (−2.51 + 15.8i)16-s + (9.72 − 9.72i)17-s + ⋯
L(s)  = 1  + (0.418 − 0.908i)2-s + (0.440 − 0.897i)3-s + (−0.649 − 0.760i)4-s + (0.642 + 0.765i)5-s + (−0.630 − 0.776i)6-s + (−0.506 + 0.506i)7-s + (−0.962 + 0.270i)8-s + (−0.611 − 0.791i)9-s + (0.964 − 0.263i)10-s + 1.53·11-s + (−0.968 + 0.247i)12-s + (−0.665 + 0.665i)13-s + (0.247 + 0.671i)14-s + (0.970 − 0.239i)15-s + (−0.157 + 0.987i)16-s + (0.572 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0687 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0687 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0687 + 0.997i$
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: \((2,\ 60,\ (\ :1),\ -0.0687 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07248 - 1.14893i\)
\(L(\frac12)\) \(\approx\) \(1.07248 - 1.14893i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.837 + 1.81i)T \)
3 \( 1 + (-1.32 + 2.69i)T \)
5 \( 1 + (-3.21 - 3.82i)T \)
good7 \( 1 + (3.54 - 3.54i)T - 49iT^{2} \)
11 \( 1 - 16.8T + 121T^{2} \)
13 \( 1 + (8.64 - 8.64i)T - 169iT^{2} \)
17 \( 1 + (-9.72 + 9.72i)T - 289iT^{2} \)
19 \( 1 - 4.78T + 361T^{2} \)
23 \( 1 + (13.5 - 13.5i)T - 529iT^{2} \)
29 \( 1 + 14.8T + 841T^{2} \)
31 \( 1 + 14.0iT - 961T^{2} \)
37 \( 1 + (10.1 + 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 6.08iT - 1.68e3T^{2} \)
43 \( 1 + (57.2 + 57.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (17.6 + 17.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-16.2 - 16.2i)T + 2.80e3iT^{2} \)
59 \( 1 + 4.37iT - 3.48e3T^{2} \)
61 \( 1 - 8.52T + 3.72e3T^{2} \)
67 \( 1 + (53.9 - 53.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 36.6T + 5.04e3T^{2} \)
73 \( 1 + (12.6 - 12.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 88.4T + 6.24e3T^{2} \)
83 \( 1 + (-63.7 + 63.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 115.T + 7.92e3T^{2} \)
97 \( 1 + (85.3 + 85.3i)T + 9.40e3iT^{2} \)
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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

−14.22430218236496448422419476023, −13.53423920073651029181817184092, −12.18635071823019405495786653212, −11.58226919852652542146504094746, −9.783941343065211241051885647944, −9.093470736434898340935473478684, −6.97619411548678425456534992260, −5.85550386348775688140689932559, −3.43289625992965186295910942184, −1.95526624958115015050565574674, 3.64177678465861949930589487293, 4.95015625249911663023825267183, 6.33569806412756792834070706188, 8.083628603583834407315493764000, 9.243756402905573429164994054789, 10.04813036443393717087554392040, 12.10078116990800685742012411989, 13.26055561788434290808345204911, 14.25437105223849669082546487374, 14.98792025702896492869007963882

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