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Properties

Label 2-60-12.11-c3-0-23
Degree $2$
Conductor $60$
Sign $-0.998 - 0.0578i$
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  + (0.622 − 2.75i)2-s + (−1.95 − 4.81i)3-s + (−7.22 − 3.43i)4-s + 5i·5-s + (−14.4 + 2.39i)6-s − 20.3i·7-s + (−13.9 + 17.7i)8-s + (−19.3 + 18.8i)9-s + (13.7 + 3.11i)10-s − 15.2·11-s + (−2.40 + 41.4i)12-s + 27.7·13-s + (−56.1 − 12.6i)14-s + (24.0 − 9.77i)15-s + (40.4 + 49.6i)16-s − 92.5i·17-s + ⋯
L(s)  = 1  + (0.220 − 0.975i)2-s + (−0.376 − 0.926i)3-s + (−0.903 − 0.429i)4-s + 0.447i·5-s + (−0.986 + 0.163i)6-s − 1.09i·7-s + (−0.617 + 0.786i)8-s + (−0.716 + 0.697i)9-s + (0.436 + 0.0984i)10-s − 0.419·11-s + (−0.0578 + 0.998i)12-s + 0.591·13-s + (−1.07 − 0.241i)14-s + (0.414 − 0.168i)15-s + (0.631 + 0.775i)16-s − 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.998 - 0.0578i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.998 - 0.0578i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0299325 + 1.03327i\)
\(L(\frac12)\) \(\approx\) \(0.0299325 + 1.03327i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.622 + 2.75i)T \)
3 \( 1 + (1.95 + 4.81i)T \)
5 \( 1 - 5iT \)
good7 \( 1 + 20.3iT - 343T^{2} \)
11 \( 1 + 15.2T + 1.33e3T^{2} \)
13 \( 1 - 27.7T + 2.19e3T^{2} \)
17 \( 1 + 92.5iT - 4.91e3T^{2} \)
19 \( 1 + 127. iT - 6.85e3T^{2} \)
23 \( 1 - 51.1T + 1.21e4T^{2} \)
29 \( 1 + 99.2iT - 2.43e4T^{2} \)
31 \( 1 + 25.8iT - 2.97e4T^{2} \)
37 \( 1 - 356.T + 5.06e4T^{2} \)
41 \( 1 - 292. iT - 6.89e4T^{2} \)
43 \( 1 - 521. iT - 7.95e4T^{2} \)
47 \( 1 + 573.T + 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 - 295.T + 2.05e5T^{2} \)
61 \( 1 - 326.T + 2.26e5T^{2} \)
67 \( 1 + 299. iT - 3.00e5T^{2} \)
71 \( 1 + 653.T + 3.57e5T^{2} \)
73 \( 1 - 504.T + 3.89e5T^{2} \)
79 \( 1 - 110. iT - 4.93e5T^{2} \)
83 \( 1 - 1.47e3T + 5.71e5T^{2} \)
89 \( 1 - 772. iT - 7.04e5T^{2} \)
97 \( 1 + 26.1T + 9.12e5T^{2} \)
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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

−13.42572580316877394882399989106, −13.20311272622801987712112840264, −11.46029141292181729609271421339, −11.05702984610228298717049125605, −9.634726215538468689231461087688, −7.902623452485460294309359066291, −6.56708573748516299997210841478, −4.81098877791758578433119661944, −2.81245786014547006769249703361, −0.74152264348654388130164651475, 3.78420347620215235584754056365, 5.31833068817796178468502125852, 6.10893995033056460614645147042, 8.235258875179545312063239319917, 9.044066893816807119538915076977, 10.36314962857268960401427007427, 12.00376326890504176999713597635, 12.93246328502256198975503645523, 14.50726943306030497453739859400, 15.27329860590427318925572536748

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