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Properties

Label 2-60-12.11-c5-0-11
Degree $2$
Conductor $60$
Sign $0.833 + 0.551i$
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  + (−4.48 − 3.45i)2-s + (−10.3 − 11.6i)3-s + (8.18 + 30.9i)4-s − 25i·5-s + (6.29 + 87.9i)6-s + 180. i·7-s + (70.0 − 166. i)8-s + (−28.0 + 241. i)9-s + (−86.2 + 112. i)10-s − 274.·11-s + (275. − 415. i)12-s + 983.·13-s + (622. − 808. i)14-s + (−291. + 259. i)15-s + (−890. + 506. i)16-s − 225. i·17-s + ⋯
L(s)  = 1  + (−0.792 − 0.610i)2-s + (−0.664 − 0.746i)3-s + (0.255 + 0.966i)4-s − 0.447i·5-s + (0.0713 + 0.997i)6-s + 1.39i·7-s + (0.387 − 0.922i)8-s + (−0.115 + 0.993i)9-s + (−0.272 + 0.354i)10-s − 0.682·11-s + (0.551 − 0.833i)12-s + 1.61·13-s + (0.848 − 1.10i)14-s + (−0.334 + 0.297i)15-s + (−0.869 + 0.494i)16-s − 0.189i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.833 + 0.551i$
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: \((2,\ 60,\ (\ :5/2),\ 0.833 + 0.551i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.846770 - 0.254847i\)
\(L(\frac12)\) \(\approx\) \(0.846770 - 0.254847i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.48 + 3.45i)T \)
3 \( 1 + (10.3 + 11.6i)T \)
5 \( 1 + 25iT \)
good7 \( 1 - 180. iT - 1.68e4T^{2} \)
11 \( 1 + 274.T + 1.61e5T^{2} \)
13 \( 1 - 983.T + 3.71e5T^{2} \)
17 \( 1 + 225. iT - 1.41e6T^{2} \)
19 \( 1 + 2.11e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.88e3T + 6.43e6T^{2} \)
29 \( 1 - 2.81e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.99e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.76e3T + 6.93e7T^{2} \)
41 \( 1 - 1.47e4iT - 1.15e8T^{2} \)
43 \( 1 + 8.63e3iT - 1.47e8T^{2} \)
47 \( 1 - 5.28e3T + 2.29e8T^{2} \)
53 \( 1 - 1.41e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.29e4T + 7.14e8T^{2} \)
61 \( 1 - 5.06e4T + 8.44e8T^{2} \)
67 \( 1 - 642. iT - 1.35e9T^{2} \)
71 \( 1 - 3.85e4T + 1.80e9T^{2} \)
73 \( 1 - 4.20e4T + 2.07e9T^{2} \)
79 \( 1 - 1.90e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.80e4T + 3.93e9T^{2} \)
89 \( 1 + 6.35e4iT - 5.58e9T^{2} \)
97 \( 1 + 4.78e4T + 8.58e9T^{2} \)
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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

−13.30062180949272706966832402719, −12.73359029843581982631664094976, −11.56631730505387910968186004269, −10.86614164287504446572822625273, −9.072662877633570725206215881145, −8.305226434065815516732796696673, −6.74086471664789793382787763168, −5.23503017360918355360419735940, −2.66389379841854783701990119001, −1.03316373992490499612791284103, 0.814402507673302769976400741834, 3.92595263415971068345234938966, 5.67585713247546225842554582825, 6.83693015075532022397509421160, 8.181344824991740159880434761010, 9.747240764571943123813204082128, 10.66651661599006189578830239899, 11.19377699887943809544596147372, 13.32147004798241503221608430917, 14.52233477405834906722578273002

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