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Properties

Label 2-60-20.7-c3-0-14
Degree $2$
Conductor $60$
Sign $0.987 + 0.154i$
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  + (2.79 − 0.431i)2-s + (2.12 + 2.12i)3-s + (7.62 − 2.40i)4-s + (2.54 − 10.8i)5-s + (6.84 + 5.01i)6-s + (−9.77 + 9.77i)7-s + (20.2 − 10.0i)8-s + 8.99i·9-s + (2.42 − 31.5i)10-s + 32.5i·11-s + (21.2 + 11.0i)12-s + (−5.95 + 5.95i)13-s + (−23.1 + 31.5i)14-s + (28.4 − 17.6i)15-s + (52.3 − 36.7i)16-s + (−71.0 − 71.0i)17-s + ⋯
L(s)  = 1  + (0.988 − 0.152i)2-s + (0.408 + 0.408i)3-s + (0.953 − 0.301i)4-s + (0.227 − 0.973i)5-s + (0.465 + 0.341i)6-s + (−0.527 + 0.527i)7-s + (0.896 − 0.443i)8-s + 0.333i·9-s + (0.0766 − 0.997i)10-s + 0.892i·11-s + (0.512 + 0.266i)12-s + (−0.127 + 0.127i)13-s + (−0.441 + 0.602i)14-s + (0.490 − 0.304i)15-s + (0.818 − 0.574i)16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.987 + 0.154i$
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: \((2,\ 60,\ (\ :3/2),\ 0.987 + 0.154i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.65627 - 0.207054i\)
\(L(\frac12)\) \(\approx\) \(2.65627 - 0.207054i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.79 + 0.431i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 + (-2.54 + 10.8i)T \)
good7 \( 1 + (9.77 - 9.77i)T - 343iT^{2} \)
11 \( 1 - 32.5iT - 1.33e3T^{2} \)
13 \( 1 + (5.95 - 5.95i)T - 2.19e3iT^{2} \)
17 \( 1 + (71.0 + 71.0i)T + 4.91e3iT^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 + (-83.3 - 83.3i)T + 1.21e4iT^{2} \)
29 \( 1 - 171. iT - 2.43e4T^{2} \)
31 \( 1 + 123. iT - 2.97e4T^{2} \)
37 \( 1 + (52.1 + 52.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 471.T + 6.89e4T^{2} \)
43 \( 1 + (258. + 258. i)T + 7.95e4iT^{2} \)
47 \( 1 + (54.3 - 54.3i)T - 1.03e5iT^{2} \)
53 \( 1 + (-331. + 331. i)T - 1.48e5iT^{2} \)
59 \( 1 - 567.T + 2.05e5T^{2} \)
61 \( 1 - 832.T + 2.26e5T^{2} \)
67 \( 1 + (-38.6 + 38.6i)T - 3.00e5iT^{2} \)
71 \( 1 - 534. iT - 3.57e5T^{2} \)
73 \( 1 + (418. - 418. i)T - 3.89e5iT^{2} \)
79 \( 1 - 76.6T + 4.93e5T^{2} \)
83 \( 1 + (908. + 908. i)T + 5.71e5iT^{2} \)
89 \( 1 + 12.1iT - 7.04e5T^{2} \)
97 \( 1 + (613. + 613. i)T + 9.12e5iT^{2} \)
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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

−14.56870847798656825728875093996, −13.23105567289988523793418588068, −12.67093881951988674039093163017, −11.40426679316922543500928579078, −9.887826544405922552020207154384, −8.853096290698865545347746005568, −6.99934035662244570307447725503, −5.36915831087548104287560910886, −4.24295632777469664775697295112, −2.31334265632951635294005347272, 2.54485348113474101634046496686, 3.93764471116883995883718978887, 6.16583087579632836203233347616, 6.89073038033552305186992248882, 8.370989113870186485107720250025, 10.38141804185029394441524000318, 11.24064733344836718030219007737, 12.83919481738421516072880798702, 13.47968393996850475703347078559, 14.53014759388547468109573871742

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