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Properties

Label 2-60-60.23-c2-0-8
Degree $2$
Conductor $60$
Sign $0.978 - 0.205i$
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 + (2.99 + 0.130i)3-s + (1.67 − 3.63i)4-s + (1.65 − 4.71i)5-s + (−5.18 + 3.01i)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 + (5.47 − 10.6i)12-s + (12.2 + 12.2i)13-s + (−5.29 − 1.15i)14-s + (5.56 − 13.9i)15-s + (−10.4 − 12.1i)16-s + (−9.47 − 9.47i)17-s + ⋯
L(s)  = 1  + (−0.841 + 0.539i)2-s + (0.999 + 0.0434i)3-s + (0.417 − 0.908i)4-s + (0.330 − 0.943i)5-s + (−0.864 + 0.502i)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 + (0.456 − 0.889i)12-s + (0.944 + 0.944i)13-s + (−0.378 − 0.0826i)14-s + (0.370 − 0.928i)15-s + (−0.651 − 0.758i)16-s + (−0.557 − 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.978 - 0.205i$
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: \((2,\ 60,\ (\ :1),\ 0.978 - 0.205i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.14103 + 0.118228i\)
\(L(\frac12)\) \(\approx\) \(1.14103 + 0.118228i\)
\(L(2)\) 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 \( 1 + (-2.99 - 0.130i)T \)
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

−15.11140683021872091503625536386, −13.90846043325363043458435636321, −12.96797820501272458391901599654, −11.19871762084009466061685420210, −9.764589059289467394349173540153, −8.793903397765263894508940890298, −8.157530955157676754822166773628, −6.50836455113144125182528094801, −4.72007222056042648736427624588, −1.91927911269681860506208875429, 2.22550409771318798866224811110, 3.67882802171696718760980827350, 6.65432020312627204131490781932, 7.938497764914361642552574626652, 8.852918828368764043224136135430, 10.45538696742109065353623498250, 10.74175405645457451703491099934, 12.71706557180198029832576762640, 13.54113781929348222552115520662, 14.92754831838976048504911242770

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