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Properties

Label 2-60-5.3-c4-0-1
Degree $2$
Conductor $60$
Sign $0.757 - 0.652i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.67 − 3.67i)3-s + (11.5 + 22.1i)5-s + (−9.71 + 9.71i)7-s + 27i·9-s + 187.·11-s + (217. + 217. i)13-s + (39.1 − 123. i)15-s + (−136. + 136. i)17-s + 153. i·19-s + 71.3·21-s + (−265. − 265. i)23-s + (−359. + 511. i)25-s + (99.2 − 99.2i)27-s − 1.21e3i·29-s + 819.·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.461 + 0.887i)5-s + (−0.198 + 0.198i)7-s + 0.333i·9-s + 1.54·11-s + (1.28 + 1.28i)13-s + (0.173 − 0.550i)15-s + (−0.471 + 0.471i)17-s + 0.423i·19-s + 0.161·21-s + (−0.502 − 0.502i)23-s + (−0.574 + 0.818i)25-s + (0.136 − 0.136i)27-s − 1.44i·29-s + 0.853·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.757 - 0.652i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.42737 + 0.530163i\)
\(L(\frac12)\) \(\approx\) \(1.42737 + 0.530163i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (3.67 + 3.67i)T \)
5 \( 1 + (-11.5 - 22.1i)T \)
good7 \( 1 + (9.71 - 9.71i)T - 2.40e3iT^{2} \)
11 \( 1 - 187.T + 1.46e4T^{2} \)
13 \( 1 + (-217. - 217. i)T + 2.85e4iT^{2} \)
17 \( 1 + (136. - 136. i)T - 8.35e4iT^{2} \)
19 \( 1 - 153. iT - 1.30e5T^{2} \)
23 \( 1 + (265. + 265. i)T + 2.79e5iT^{2} \)
29 \( 1 + 1.21e3iT - 7.07e5T^{2} \)
31 \( 1 - 819.T + 9.23e5T^{2} \)
37 \( 1 + (1.48e3 - 1.48e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 233.T + 2.82e6T^{2} \)
43 \( 1 + (1.12e3 + 1.12e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.04e3 + 1.04e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (995. + 995. i)T + 7.89e6iT^{2} \)
59 \( 1 + 2.85e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.33e3T + 1.38e7T^{2} \)
67 \( 1 + (1.19e3 - 1.19e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 8.81e3T + 2.54e7T^{2} \)
73 \( 1 + (326. + 326. i)T + 2.83e7iT^{2} \)
79 \( 1 + 6.73e3iT - 3.89e7T^{2} \)
83 \( 1 + (334. + 334. i)T + 4.74e7iT^{2} \)
89 \( 1 + 3.12e3iT - 6.27e7T^{2} \)
97 \( 1 + (-5.29e3 + 5.29e3i)T - 8.85e7iT^{2} \)
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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

−14.21754968159875593965552465279, −13.57459866148073941154806024045, −11.98451717152352916072199537803, −11.23165359405389865821212644568, −9.889968202101867352030224753547, −8.578680029019545925793716574090, −6.65143224063657351315039671779, −6.23512163985146464946271264412, −3.92026165563827639042486941628, −1.76821654879350274912546019056, 1.04498397352996026039602566346, 3.79584262259532360228279914938, 5.33492189498502780143639211037, 6.56779360136946534299536410585, 8.543253768761890105209046251854, 9.475062250501882772930829875995, 10.75174950768686696970575211723, 11.95540239141758919455664190873, 13.07660568831879886874281452402, 14.09263206573185394164110298142

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