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Properties

Label 2-60-12.11-c5-0-20
Degree $2$
Conductor $60$
Sign $0.657 - 0.753i$
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  + (−2.24 + 5.19i)2-s + (15.5 − 1.52i)3-s + (−21.9 − 23.2i)4-s − 25i·5-s + (−26.8 + 83.9i)6-s + 85.2i·7-s + (170. − 61.6i)8-s + (238. − 47.3i)9-s + (129. + 56.0i)10-s + 246.·11-s + (−375. − 327. i)12-s + 940.·13-s + (−442. − 191. i)14-s + (−38.1 − 387. i)15-s + (−61.4 + 1.02e3i)16-s − 639. i·17-s + ⋯
L(s)  = 1  + (−0.396 + 0.918i)2-s + (0.995 − 0.0979i)3-s + (−0.685 − 0.728i)4-s − 0.447i·5-s + (−0.304 + 0.952i)6-s + 0.657i·7-s + (0.940 − 0.340i)8-s + (0.980 − 0.194i)9-s + (0.410 + 0.177i)10-s + 0.613·11-s + (−0.753 − 0.657i)12-s + 1.54·13-s + (−0.603 − 0.260i)14-s + (−0.0437 − 0.445i)15-s + (−0.0600 + 0.998i)16-s − 0.536i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.657 - 0.753i$
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: \((2,\ 60,\ (\ :5/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.82108 + 0.827971i\)
\(L(\frac12)\) \(\approx\) \(1.82108 + 0.827971i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.24 - 5.19i)T \)
3 \( 1 + (-15.5 + 1.52i)T \)
5 \( 1 + 25iT \)
good7 \( 1 - 85.2iT - 1.68e4T^{2} \)
11 \( 1 - 246.T + 1.61e5T^{2} \)
13 \( 1 - 940.T + 3.71e5T^{2} \)
17 \( 1 + 639. iT - 1.41e6T^{2} \)
19 \( 1 - 2.05e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.54e3T + 6.43e6T^{2} \)
29 \( 1 + 2.38e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.33e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.37e4T + 6.93e7T^{2} \)
41 \( 1 + 1.92e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.26e4iT - 1.47e8T^{2} \)
47 \( 1 - 5.17e3T + 2.29e8T^{2} \)
53 \( 1 - 1.80e4iT - 4.18e8T^{2} \)
59 \( 1 - 9.34e3T + 7.14e8T^{2} \)
61 \( 1 - 4.68e3T + 8.44e8T^{2} \)
67 \( 1 - 1.96e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.61e4T + 1.80e9T^{2} \)
73 \( 1 + 5.44e4T + 2.07e9T^{2} \)
79 \( 1 - 2.81e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.17e5T + 3.93e9T^{2} \)
89 \( 1 + 6.72e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.53e3T + 8.58e9T^{2} \)
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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

−14.28173630785264993375513909469, −13.56736921137671378721534648940, −12.25643695001296342397057078637, −10.34681250397370284892377569334, −8.922796886078919997691016956540, −8.593483844001018498753139502327, −7.11548493614030030840220340785, −5.67735440196770180724892404130, −3.87129237785780564051993086317, −1.42474579912690149131834512241, 1.34884525906048626798111469249, 3.12600189070035875247228026943, 4.20892006530762829698966629905, 6.94425220746445314974170086246, 8.317047913112503654058621853775, 9.260247418252852343933853031825, 10.48637480324665956384114978544, 11.36032278111726853751223445164, 13.06917326490498955282340087082, 13.64618448066455716323033873036

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