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Properties

Label 2-60-4.3-c4-0-12
Degree $2$
Conductor $60$
Sign $-0.981 + 0.192i$
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  + (−3.08 − 2.54i)2-s − 5.19i·3-s + (3.07 + 15.7i)4-s + 11.1·5-s + (−13.2 + 16.0i)6-s − 86.5i·7-s + (30.4 − 56.3i)8-s − 27·9-s + (−34.5 − 28.4i)10-s + 97.6i·11-s + (81.5 − 15.9i)12-s − 297.·13-s + (−220. + 267. i)14-s − 58.0i·15-s + (−237. + 96.6i)16-s − 58.8·17-s + ⋯
L(s)  = 1  + (−0.772 − 0.635i)2-s − 0.577i·3-s + (0.192 + 0.981i)4-s + 0.447·5-s + (−0.366 + 0.445i)6-s − 1.76i·7-s + (0.475 − 0.879i)8-s − 0.333·9-s + (−0.345 − 0.284i)10-s + 0.807i·11-s + (0.566 − 0.111i)12-s − 1.76·13-s + (−1.12 + 1.36i)14-s − 0.258i·15-s + (−0.926 + 0.377i)16-s − 0.203·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.981 + 0.192i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.981 + 0.192i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0753009 - 0.776008i\)
\(L(\frac12)\) \(\approx\) \(0.0753009 - 0.776008i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.08 + 2.54i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 - 11.1T \)
good7 \( 1 + 86.5iT - 2.40e3T^{2} \)
11 \( 1 - 97.6iT - 1.46e4T^{2} \)
13 \( 1 + 297.T + 2.85e4T^{2} \)
17 \( 1 + 58.8T + 8.35e4T^{2} \)
19 \( 1 + 497. iT - 1.30e5T^{2} \)
23 \( 1 + 120. iT - 2.79e5T^{2} \)
29 \( 1 + 947.T + 7.07e5T^{2} \)
31 \( 1 + 559. iT - 9.23e5T^{2} \)
37 \( 1 - 1.16e3T + 1.87e6T^{2} \)
41 \( 1 - 3.17e3T + 2.82e6T^{2} \)
43 \( 1 + 1.16e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.06e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.44e3T + 7.89e6T^{2} \)
59 \( 1 - 2.00e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.52e3T + 1.38e7T^{2} \)
67 \( 1 - 6.16e3iT - 2.01e7T^{2} \)
71 \( 1 + 882. iT - 2.54e7T^{2} \)
73 \( 1 - 1.65e3T + 2.83e7T^{2} \)
79 \( 1 + 7.27e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.34e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.00e3T + 6.27e7T^{2} \)
97 \( 1 + 9.79e3T + 8.85e7T^{2} \)
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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

−13.44032502038702382068593003081, −12.72205969841933065466728333567, −11.36687659272996691725362504540, −10.25464116412468361259427399870, −9.371063798773337683720977657377, −7.50560324599342979115526095113, −7.06800112490537221891306160873, −4.41345806854573563814252871704, −2.33218894619593470299397897788, −0.54007617467106311631963410453, 2.37973802514890854709866316991, 5.23444812645998665533278841106, 6.05280868974063968379500264607, 7.927753675566272869065379999953, 9.140676194926141407845160482532, 9.790479371497224286317893770829, 11.22776593075950419536118165229, 12.46339780165955247347114624972, 14.37039768506942612595818681712, 14.92708067948955685077674090414

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