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Properties

Label 2-60-12.11-c5-0-16
Degree $2$
Conductor $60$
Sign $0.850 - 0.525i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.00749 + 5.65i)2-s + (−8.15 − 13.2i)3-s + (−31.9 + 0.0848i)4-s + 25i·5-s + (75.0 − 46.2i)6-s − 55.6i·7-s + (−0.719 − 181. i)8-s + (−109. + 216. i)9-s + (−141. + 0.187i)10-s + 664.·11-s + (262. + 424. i)12-s + 725.·13-s + (314. − 0.416i)14-s + (332. − 203. i)15-s + (1.02e3 − 5.42i)16-s − 42.4i·17-s + ⋯
L(s)  = 1  + (0.00132 + 0.999i)2-s + (−0.523 − 0.852i)3-s + (−0.999 + 0.00265i)4-s + 0.447i·5-s + (0.851 − 0.524i)6-s − 0.428i·7-s + (−0.00397 − 0.999i)8-s + (−0.452 + 0.891i)9-s + (−0.447 + 0.000592i)10-s + 1.65·11-s + (0.525 + 0.850i)12-s + 1.19·13-s + (0.428 − 0.000568i)14-s + (0.381 − 0.233i)15-s + (0.999 − 0.00530i)16-s − 0.0356i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.33337 + 0.378551i\)
\(L(\frac12)\) \(\approx\) \(1.33337 + 0.378551i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.00749 - 5.65i)T \)
3 \( 1 + (8.15 + 13.2i)T \)
5 \( 1 - 25iT \)
good7 \( 1 + 55.6iT - 1.68e4T^{2} \)
11 \( 1 - 664.T + 1.61e5T^{2} \)
13 \( 1 - 725.T + 3.71e5T^{2} \)
17 \( 1 + 42.4iT - 1.41e6T^{2} \)
19 \( 1 - 829. iT - 2.47e6T^{2} \)
23 \( 1 - 1.46e3T + 6.43e6T^{2} \)
29 \( 1 + 365. iT - 2.05e7T^{2} \)
31 \( 1 + 7.94e3iT - 2.86e7T^{2} \)
37 \( 1 - 7.60e3T + 6.93e7T^{2} \)
41 \( 1 - 1.16e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.13e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.09e4T + 2.29e8T^{2} \)
53 \( 1 - 1.76e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.60e4T + 7.14e8T^{2} \)
61 \( 1 + 3.94e4T + 8.44e8T^{2} \)
67 \( 1 + 2.16e3iT - 1.35e9T^{2} \)
71 \( 1 + 1.61e4T + 1.80e9T^{2} \)
73 \( 1 - 6.86e4T + 2.07e9T^{2} \)
79 \( 1 - 1.04e5iT - 3.07e9T^{2} \)
83 \( 1 - 9.25e4T + 3.93e9T^{2} \)
89 \( 1 - 1.63e3iT - 5.58e9T^{2} \)
97 \( 1 - 4.72e4T + 8.58e9T^{2} \)
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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

−14.08482766577237359859979412818, −13.41386333001684435844808702498, −12.06780702235692892751168551519, −10.84592090393102818482397868212, −9.195578564643310800534545484599, −7.84737394692246510292321383127, −6.70470988560997204498583143184, −5.93964579994513356962641112952, −3.99746439192005057757465385313, −1.03969252549329474057421848777, 1.09916231903874667232048412664, 3.49945936274393097167131999728, 4.69038050932197204700078657549, 6.14156097907389250443369476774, 8.800106906729685070361486110144, 9.291425791972529346701404486538, 10.74677954751345335976720028347, 11.60234333147117323284117597143, 12.48541675829825789051402598350, 13.88995640998457133445027566904

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