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Properties

Label 2-60-12.11-c5-0-29
Degree $2$
Conductor $60$
Sign $0.181 + 0.983i$
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  + (−5.54 + 1.12i)2-s + (15.2 + 3.36i)3-s + (29.4 − 12.4i)4-s − 25i·5-s + (−88.1 − 1.54i)6-s − 209. i·7-s + (−149. + 102. i)8-s + (220. + 102. i)9-s + (28.1 + 138. i)10-s − 618.·11-s + (490. − 90.6i)12-s − 94.2·13-s + (235. + 1.16e3i)14-s + (84.1 − 380. i)15-s + (712. − 735. i)16-s − 333. i·17-s + ⋯
L(s)  = 1  + (−0.980 + 0.198i)2-s + (0.976 + 0.216i)3-s + (0.920 − 0.389i)4-s − 0.447i·5-s + (−0.999 − 0.0175i)6-s − 1.61i·7-s + (−0.825 + 0.565i)8-s + (0.906 + 0.421i)9-s + (0.0889 + 0.438i)10-s − 1.54·11-s + (0.983 − 0.181i)12-s − 0.154·13-s + (0.321 + 1.58i)14-s + (0.0965 − 0.436i)15-s + (0.696 − 0.717i)16-s − 0.279i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.181 + 0.983i$
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: \((2,\ 60,\ (\ :5/2),\ 0.181 + 0.983i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.962204 - 0.800736i\)
\(L(\frac12)\) \(\approx\) \(0.962204 - 0.800736i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.54 - 1.12i)T \)
3 \( 1 + (-15.2 - 3.36i)T \)
5 \( 1 + 25iT \)
good7 \( 1 + 209. iT - 1.68e4T^{2} \)
11 \( 1 + 618.T + 1.61e5T^{2} \)
13 \( 1 + 94.2T + 3.71e5T^{2} \)
17 \( 1 + 333. iT - 1.41e6T^{2} \)
19 \( 1 + 3.06e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.82e3T + 6.43e6T^{2} \)
29 \( 1 + 5.16e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.76e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.39e3T + 6.93e7T^{2} \)
41 \( 1 + 2.28e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.57e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.62e4T + 2.29e8T^{2} \)
53 \( 1 - 8.56e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.78e4T + 7.14e8T^{2} \)
61 \( 1 + 3.26e3T + 8.44e8T^{2} \)
67 \( 1 - 8.19e3iT - 1.35e9T^{2} \)
71 \( 1 + 4.90e4T + 1.80e9T^{2} \)
73 \( 1 - 9.21e3T + 2.07e9T^{2} \)
79 \( 1 - 5.42e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.22e4T + 3.93e9T^{2} \)
89 \( 1 + 4.04e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.17e4T + 8.58e9T^{2} \)
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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

−13.82508442941284111612523674981, −13.03796013783066485067935221155, −10.99597653797346980804516576240, −10.18772029962033033480292458297, −9.088478096601186335974267199047, −7.83805547160848342520710360425, −7.12440638481459105412823865625, −4.74627325633966523772199425624, −2.72358978622529564637375441151, −0.71003977653287971493401622926, 2.05838376635618639751028116882, 3.05796248018923889351195393740, 5.90740048545864501654056864364, 7.60375602412427508256234610109, 8.409727348010728537691763283308, 9.517573616278195892977714500594, 10.59389535217902797454068441157, 12.12452541694436799127948167148, 12.94770422456442456187484928057, 14.73967533542410339523337198465

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