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Properties

Label 2-60-20.7-c3-0-9
Degree $2$
Conductor $60$
Sign $0.378 - 0.925i$
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  + (2.57 + 1.16i)2-s + (2.12 + 2.12i)3-s + (5.29 + 6.00i)4-s + (−8.61 + 7.12i)5-s + (3 + 7.93i)6-s + (8.98 − 8.98i)7-s + (6.65 + 21.6i)8-s + 8.99i·9-s + (−30.4 + 8.35i)10-s − 38.8i·11-s + (−1.50 + 23.9i)12-s + (28.7 − 28.7i)13-s + (33.6 − 12.7i)14-s + (−33.3 − 3.14i)15-s + (−8 + 63.4i)16-s + (11.2 + 11.2i)17-s + ⋯
L(s)  = 1  + (0.911 + 0.411i)2-s + (0.408 + 0.408i)3-s + (0.661 + 0.750i)4-s + (−0.770 + 0.637i)5-s + (0.204 + 0.540i)6-s + (0.485 − 0.485i)7-s + (0.294 + 0.955i)8-s + 0.333i·9-s + (−0.964 + 0.264i)10-s − 1.06i·11-s + (−0.0361 + 0.576i)12-s + (0.613 − 0.613i)13-s + (0.641 − 0.242i)14-s + (−0.574 − 0.0541i)15-s + (−0.125 + 0.992i)16-s + (0.161 + 0.161i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.378 - 0.925i$
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: \((2,\ 60,\ (\ :3/2),\ 0.378 - 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.00791 + 1.34784i\)
\(L(\frac12)\) \(\approx\) \(2.00791 + 1.34784i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.57 - 1.16i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 + (8.61 - 7.12i)T \)
good7 \( 1 + (-8.98 + 8.98i)T - 343iT^{2} \)
11 \( 1 + 38.8iT - 1.33e3T^{2} \)
13 \( 1 + (-28.7 + 28.7i)T - 2.19e3iT^{2} \)
17 \( 1 + (-11.2 - 11.2i)T + 4.91e3iT^{2} \)
19 \( 1 - 15.7T + 6.85e3T^{2} \)
23 \( 1 + (106. + 106. i)T + 1.21e4iT^{2} \)
29 \( 1 - 208. iT - 2.43e4T^{2} \)
31 \( 1 + 243. iT - 2.97e4T^{2} \)
37 \( 1 + (203. + 203. i)T + 5.06e4iT^{2} \)
41 \( 1 - 25.7T + 6.89e4T^{2} \)
43 \( 1 + (-253. - 253. i)T + 7.95e4iT^{2} \)
47 \( 1 + (366. - 366. i)T - 1.03e5iT^{2} \)
53 \( 1 + (501. - 501. i)T - 1.48e5iT^{2} \)
59 \( 1 - 646.T + 2.05e5T^{2} \)
61 \( 1 - 527.T + 2.26e5T^{2} \)
67 \( 1 + (392. - 392. i)T - 3.00e5iT^{2} \)
71 \( 1 + 611. iT - 3.57e5T^{2} \)
73 \( 1 + (-144. + 144. i)T - 3.89e5iT^{2} \)
79 \( 1 + 551.T + 4.93e5T^{2} \)
83 \( 1 + (494. + 494. i)T + 5.71e5iT^{2} \)
89 \( 1 + 740. iT - 7.04e5T^{2} \)
97 \( 1 + (-84.7 - 84.7i)T + 9.12e5iT^{2} \)
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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

−14.57377966853072872836353297365, −14.08757666280583162559101760181, −12.76436837802215643297203715154, −11.32275334936716881992133537359, −10.65415120094852927864481795543, −8.426667243505626866739392174502, −7.57991539084789198485876912193, −6.01079043994653536258748530909, −4.26040409610262950171206221321, −3.13797332909337948928867993021, 1.77244662142129233223833622975, 3.82748145657299430699867170006, 5.18721561351192915261918101284, 6.95869045496323205334717814509, 8.318217689105691779554039549356, 9.791664720073018647146332705277, 11.57016403161697497185443435669, 12.08606938453085921069561872583, 13.19286831775159545508810186450, 14.25800284769609896759918253760

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