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Properties

Label 2-60-12.11-c5-0-0
Degree $2$
Conductor $60$
Sign $-0.932 + 0.362i$
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  + (1.26 + 5.51i)2-s + (1.23 − 15.5i)3-s + (−28.8 + 13.9i)4-s − 25i·5-s + (87.2 − 12.7i)6-s + 173. i·7-s + (−113. − 141. i)8-s + (−239. − 38.4i)9-s + (137. − 31.5i)10-s − 400.·11-s + (180. + 464. i)12-s − 895.·13-s + (−956. + 219. i)14-s + (−388. − 30.9i)15-s + (636. − 802. i)16-s − 138. i·17-s + ⋯
L(s)  = 1  + (0.223 + 0.974i)2-s + (0.0793 − 0.996i)3-s + (−0.900 + 0.434i)4-s − 0.447i·5-s + (0.989 − 0.145i)6-s + 1.33i·7-s + (−0.624 − 0.780i)8-s + (−0.987 − 0.158i)9-s + (0.435 − 0.0997i)10-s − 0.997·11-s + (0.362 + 0.932i)12-s − 1.47·13-s + (−1.30 + 0.298i)14-s + (−0.445 − 0.0354i)15-s + (0.621 − 0.783i)16-s − 0.116i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.932 + 0.362i$
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: \((2,\ 60,\ (\ :5/2),\ -0.932 + 0.362i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0306935 - 0.163774i\)
\(L(\frac12)\) \(\approx\) \(0.0306935 - 0.163774i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 5.51i)T \)
3 \( 1 + (-1.23 + 15.5i)T \)
5 \( 1 + 25iT \)
good7 \( 1 - 173. iT - 1.68e4T^{2} \)
11 \( 1 + 400.T + 1.61e5T^{2} \)
13 \( 1 + 895.T + 3.71e5T^{2} \)
17 \( 1 + 138. iT - 1.41e6T^{2} \)
19 \( 1 - 852. iT - 2.47e6T^{2} \)
23 \( 1 + 924.T + 6.43e6T^{2} \)
29 \( 1 + 8.07e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.29e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.44e3T + 6.93e7T^{2} \)
41 \( 1 - 1.37e4iT - 1.15e8T^{2} \)
43 \( 1 - 5.13e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.84e4T + 2.29e8T^{2} \)
53 \( 1 + 2.78e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.96e4T + 7.14e8T^{2} \)
61 \( 1 + 1.86e4T + 8.44e8T^{2} \)
67 \( 1 + 2.47e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.44e4T + 1.80e9T^{2} \)
73 \( 1 - 3.49e4T + 2.07e9T^{2} \)
79 \( 1 - 5.31e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.36e4T + 3.93e9T^{2} \)
89 \( 1 + 3.26e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.21e5T + 8.58e9T^{2} \)
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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

−14.76356914059241381956618796277, −13.61649367607464884248455822333, −12.53084933300137061429297119578, −12.01398926246516014135822439812, −9.623475273367152028386721772395, −8.399520475469701026887290882982, −7.57044229361899003125986365533, −6.05203348428584527370237218200, −5.07562084650860409421214006491, −2.54569619852679360986444062932, 0.06838980155138364153447623532, 2.70364541301738423176460070707, 4.09122335072133803811260006888, 5.24780233843630016001423318109, 7.49998300999293100858808616172, 9.213783438277583499801689287764, 10.50461446326133690853651037457, 10.61380573569353508611956441210, 12.18436509326338874468872937538, 13.58827126498600196511381001334

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