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Properties

Label 2-60-20.7-c3-0-1
Degree $2$
Conductor $60$
Sign $-0.943 + 0.331i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.949 + 2.66i)2-s + (2.12 + 2.12i)3-s + (−6.19 − 5.05i)4-s + (−10.6 − 3.47i)5-s + (−7.66 + 3.63i)6-s + (−24.7 + 24.7i)7-s + (19.3 − 11.7i)8-s + 8.99i·9-s + (19.3 − 25.0i)10-s − 35.1i·11-s + (−2.41 − 23.8i)12-s + (−28.9 + 28.9i)13-s + (−42.4 − 89.3i)14-s + (−15.1 − 29.9i)15-s + (12.8 + 62.7i)16-s + (54.2 + 54.2i)17-s + ⋯
L(s)  = 1  + (−0.335 + 0.941i)2-s + (0.408 + 0.408i)3-s + (−0.774 − 0.632i)4-s + (−0.950 − 0.311i)5-s + (−0.521 + 0.247i)6-s + (−1.33 + 1.33i)7-s + (0.855 − 0.517i)8-s + 0.333i·9-s + (0.612 − 0.790i)10-s − 0.963i·11-s + (−0.0580 − 0.574i)12-s + (−0.618 + 0.618i)13-s + (−0.809 − 1.70i)14-s + (−0.260 − 0.515i)15-s + (0.200 + 0.979i)16-s + (0.773 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.943 + 0.331i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0850605 - 0.498712i\)
\(L(\frac12)\) \(\approx\) \(0.0850605 - 0.498712i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.949 - 2.66i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 + (10.6 + 3.47i)T \)
good7 \( 1 + (24.7 - 24.7i)T - 343iT^{2} \)
11 \( 1 + 35.1iT - 1.33e3T^{2} \)
13 \( 1 + (28.9 - 28.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (-54.2 - 54.2i)T + 4.91e3iT^{2} \)
19 \( 1 + 20.8T + 6.85e3T^{2} \)
23 \( 1 + (14.3 + 14.3i)T + 1.21e4iT^{2} \)
29 \( 1 - 120. iT - 2.43e4T^{2} \)
31 \( 1 - 183. iT - 2.97e4T^{2} \)
37 \( 1 + (-96.2 - 96.2i)T + 5.06e4iT^{2} \)
41 \( 1 + 363.T + 6.89e4T^{2} \)
43 \( 1 + (159. + 159. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-24.6 + 24.6i)T - 1.03e5iT^{2} \)
53 \( 1 + (-270. + 270. i)T - 1.48e5iT^{2} \)
59 \( 1 - 417.T + 2.05e5T^{2} \)
61 \( 1 + 110.T + 2.26e5T^{2} \)
67 \( 1 + (-51.4 + 51.4i)T - 3.00e5iT^{2} \)
71 \( 1 + 291. iT - 3.57e5T^{2} \)
73 \( 1 + (752. - 752. i)T - 3.89e5iT^{2} \)
79 \( 1 - 127.T + 4.93e5T^{2} \)
83 \( 1 + (-187. - 187. i)T + 5.71e5iT^{2} \)
89 \( 1 - 51.0iT - 7.04e5T^{2} \)
97 \( 1 + (-676. - 676. i)T + 9.12e5iT^{2} \)
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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

−15.36934571239563006599584342297, −14.55055671083239190049612112513, −13.08300198338727964248503679796, −11.99130335831003517272883220979, −10.19306643280359379995445513093, −8.983465150929743154518117504259, −8.357543553495859857949718745940, −6.71092930367262107802874869159, −5.32251731349937575865383401276, −3.50201468941079699080448804445, 0.37450738076588771894477437074, 2.99367537696413428245035359817, 4.18049897433553886147560130208, 7.11439457908603746512381236915, 7.81798655273749011656437357266, 9.630369731946464583519954247382, 10.32469574819475368989197053066, 11.81035042055013113410471858503, 12.74428962595243090500010994544, 13.60360150595673123354175815304

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