Properties

Label 2-60-60.47-c2-0-15
Degree $2$
Conductor $60$
Sign $0.747 + 0.664i$
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  + (1.99 − 0.141i)2-s + (−2.17 − 2.06i)3-s + (3.95 − 0.565i)4-s + (3.07 − 3.94i)5-s + (−4.63 − 3.81i)6-s + (−5.18 + 5.18i)7-s + (7.81 − 1.68i)8-s + (0.459 + 8.98i)9-s + (5.57 − 8.29i)10-s + 7.14·11-s + (−9.78 − 6.95i)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  + (0.997 − 0.0708i)2-s + (−0.724 − 0.688i)3-s + (0.989 − 0.141i)4-s + (0.615 − 0.788i)5-s + (−0.771 − 0.635i)6-s + (−0.741 + 0.741i)7-s + (0.977 − 0.211i)8-s + (0.0510 + 0.998i)9-s + (0.557 − 0.829i)10-s + 0.649·11-s + (−0.815 − 0.579i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.747 + 0.664i$
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: \((2,\ 60,\ (\ :1),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61010 - 0.612548i\)
\(L(\frac12)\) \(\approx\) \(1.61010 - 0.612548i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.141i)T \)
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

−14.47619100082564049690025384306, −13.25239257339194156792654072339, −12.58580133683937317141137165560, −11.84302307279415384567335234134, −10.41892156651927349871964393396, −8.792220619346816227656934729729, −6.75927690842339263386883739526, −5.98345953247921496815715516384, −4.63453720798339912495383345240, −2.06237987368414699531691447411, 3.18593788053680329378922370223, 4.72581375215691755531891321956, 6.25339269530633731171869977223, 7.00879640381916581318760844215, 9.646399443930433570481442725127, 10.58752263089231875692567969972, 11.52547537822386809580406986089, 12.85572974867889146525694648286, 13.91196342376524954298778132307, 14.95400330371349492292416086073

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