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Properties

Label 2-60-60.23-c2-0-6
Degree $2$
Conductor $60$
Sign $0.531 - 0.847i$
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  + (0.141 + 1.99i)2-s + (2.17 − 2.06i)3-s + (−3.95 + 0.565i)4-s + (3.07 + 3.94i)5-s + (4.43 + 4.04i)6-s + (5.18 + 5.18i)7-s + (−1.68 − 7.81i)8-s + (0.459 − 8.98i)9-s + (−7.42 + 6.69i)10-s − 7.14·11-s + (−7.44 + 9.41i)12-s + (−7.93 − 7.93i)13-s + (−9.61 + 11.0i)14-s + (14.8 + 2.21i)15-s + (15.3 − 4.47i)16-s + (−16.5 − 16.5i)17-s + ⋯
L(s)  = 1  + (0.0708 + 0.997i)2-s + (0.724 − 0.688i)3-s + (−0.989 + 0.141i)4-s + (0.615 + 0.788i)5-s + (0.738 + 0.674i)6-s + (0.741 + 0.741i)7-s + (−0.211 − 0.977i)8-s + (0.0510 − 0.998i)9-s + (−0.742 + 0.669i)10-s − 0.649·11-s + (−0.620 + 0.784i)12-s + (−0.610 − 0.610i)13-s + (−0.686 + 0.791i)14-s + (0.989 + 0.147i)15-s + (0.960 − 0.279i)16-s + (−0.975 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.531 - 0.847i$
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: \((2,\ 60,\ (\ :1),\ 0.531 - 0.847i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.28381 + 0.710391i\)
\(L(\frac12)\) \(\approx\) \(1.28381 + 0.710391i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.141 - 1.99i)T \)
3 \( 1 + (-2.17 + 2.06i)T \)
5 \( 1 + (-3.07 - 3.94i)T \)
good7 \( 1 + (-5.18 - 5.18i)T + 49iT^{2} \)
11 \( 1 + 7.14T + 121T^{2} \)
13 \( 1 + (7.93 + 7.93i)T + 169iT^{2} \)
17 \( 1 + (16.5 + 16.5i)T + 289iT^{2} \)
19 \( 1 - 12.1T + 361T^{2} \)
23 \( 1 + (11.0 + 11.0i)T + 529iT^{2} \)
29 \( 1 - 26.1T + 841T^{2} \)
31 \( 1 - 8.74iT - 961T^{2} \)
37 \( 1 + (26.7 - 26.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 35.4iT - 1.68e3T^{2} \)
43 \( 1 + (24.6 - 24.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-58.6 + 58.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (20.4 - 20.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 59.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.42T + 3.72e3T^{2} \)
67 \( 1 + (-35.8 - 35.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 46.2T + 5.04e3T^{2} \)
73 \( 1 + (-10.6 - 10.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 68.3T + 6.24e3T^{2} \)
83 \( 1 + (76.6 + 76.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 41.0T + 7.92e3T^{2} \)
97 \( 1 + (-81.7 + 81.7i)T - 9.40e3iT^{2} \)
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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

−14.93207002769203091593106527738, −14.06057863802021776466438219068, −13.27850474370611357578409614861, −11.95282596025989094844925014140, −10.05266935968161128578412738083, −8.800593939262688116564533674906, −7.71696812605911348286856428657, −6.62739007794500938117237317989, −5.19602866485471054584763736544, −2.70261779211372574898093415116, 2.03530202940777314142695447258, 4.13246535844901033221061940800, 5.14397653881012939447473899730, 7.988842898895430713286623764743, 9.057114804254575407749743002693, 10.08242900564529971976497368781, 10.99489253539164848574304023734, 12.51600739935217230129244192918, 13.71021254376755572586715146569, 14.16450135218973533264382290116

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