Properties

Label 2-60-20.19-c2-0-9
Degree $2$
Conductor $60$
Sign $0.375 + 0.926i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52 − 1.29i)2-s − 1.73·3-s + (0.637 − 3.94i)4-s + (4.27 − 2.59i)5-s + (−2.63 + 2.24i)6-s − 0.837·7-s + (−4.14 − 6.83i)8-s + 2.99·9-s + (3.14 − 9.49i)10-s + 15.7i·11-s + (−1.10 + 6.83i)12-s + 5.18i·13-s + (−1.27 + 1.08i)14-s + (−7.40 + 4.49i)15-s + (−15.1 − 5.03i)16-s + 27.3i·17-s + ⋯
L(s)  = 1  + (0.761 − 0.648i)2-s − 0.577·3-s + (0.159 − 0.987i)4-s + (0.854 − 0.518i)5-s + (−0.439 + 0.374i)6-s − 0.119·7-s + (−0.518 − 0.854i)8-s + 0.333·9-s + (0.314 − 0.949i)10-s + 1.43i·11-s + (−0.0920 + 0.569i)12-s + 0.398i·13-s + (−0.0910 + 0.0775i)14-s + (−0.493 + 0.299i)15-s + (−0.949 − 0.314i)16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.375 + 0.926i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.375 + 0.926i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30648 - 0.880039i\)
\(L(\frac12)\) \(\approx\) \(1.30648 - 0.880039i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.52 + 1.29i)T \)
3 \( 1 + 1.73T \)
5 \( 1 + (-4.27 + 2.59i)T \)
good7 \( 1 + 0.837T + 49T^{2} \)
11 \( 1 - 15.7iT - 121T^{2} \)
13 \( 1 - 5.18iT - 169T^{2} \)
17 \( 1 - 27.3iT - 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 - 19.1T + 529T^{2} \)
29 \( 1 + 45.6T + 841T^{2} \)
31 \( 1 + 13.6iT - 961T^{2} \)
37 \( 1 + 15.5iT - 1.36e3T^{2} \)
41 \( 1 - 13.2T + 1.68e3T^{2} \)
43 \( 1 + 27.9T + 1.84e3T^{2} \)
47 \( 1 - 55.6T + 2.20e3T^{2} \)
53 \( 1 + 15.5iT - 2.80e3T^{2} \)
59 \( 1 + 87.6iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 + 92.2T + 4.48e3T^{2} \)
71 \( 1 - 130. iT - 5.04e3T^{2} \)
73 \( 1 - 54.7iT - 5.32e3T^{2} \)
79 \( 1 - 13.6iT - 6.24e3T^{2} \)
83 \( 1 + 59.0T + 6.88e3T^{2} \)
89 \( 1 - 39.8T + 7.92e3T^{2} \)
97 \( 1 + 168. iT - 9.40e3T^{2} \)
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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

−14.54100323514059263677162290494, −13.01730386432228907246064621414, −12.73632039353432104826516018771, −11.33575666050919452642758535775, −10.18382102111319650356110353220, −9.240950070575764606447463136062, −6.83431869619299086821250699683, −5.56387263243683371047532774365, −4.35863472850823854302830513567, −1.86973123275540396521949781418, 3.16758847432878325446883541284, 5.29072090516213096373086466641, 6.16088186568913454431046374600, 7.42764247586753461063307896674, 9.123223274633683100827855288184, 10.73934142512084256784712791096, 11.76148279849183001497612579135, 13.17056814466844008011466262116, 13.87202210096567376554198352865, 14.94463804848334619545802800367

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