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Properties

Label 2-60-20.7-c3-0-2
Degree $2$
Conductor $60$
Sign $-0.965 - 0.260i$
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  + (1.16 + 2.57i)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 + (−21.6 − 6.65i)8-s + 8.99i·9-s + (−28.4 − 13.9i)10-s + 38.8i·11-s + (23.9 − 1.50i)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  + (0.411 + 0.911i)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 + (−0.955 − 0.294i)8-s + 0.333i·9-s + (−0.898 − 0.439i)10-s + 1.06i·11-s + (0.576 − 0.0361i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.965 - 0.260i$
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: \((2,\ 60,\ (\ :3/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.116646 + 0.881550i\)
\(L(\frac12)\) \(\approx\) \(0.116646 + 0.881550i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 2.57i)T \)
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

−15.33247750651530985998395343837, −14.15016764704286783576133481154, −12.76988494792564319131157645194, −12.14902613373558024918480879038, −10.64749054941376201463010154749, −8.898837129681861138843753022661, −7.50210773387602190530059793080, −6.66712366698615439589719467097, −5.22121415281928660739408956333, −3.39989993554528528026167334904, 0.59197363735941072899412914189, 3.51048371246012241795987612224, 4.65240302903879204276702561963, 6.23976906984149937513327211514, 8.434820824260373714018494801419, 9.608083196930124868162340874987, 10.96688260716898894893551953031, 11.63384602205521827921690848956, 12.85973483775912935876702384010, 13.76496527390342933279099969498

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