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Properties

Label 2-60-12.11-c5-0-30
Degree $2$
Conductor $60$
Sign $-0.221 + 0.975i$
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  + (−3.19 + 4.66i)2-s + (8.70 − 12.9i)3-s + (−11.5 − 29.8i)4-s + 25i·5-s + (32.5 + 81.9i)6-s − 40.1i·7-s + (176. + 41.5i)8-s + (−91.5 − 225. i)9-s + (−116. − 79.9i)10-s − 401.·11-s + (−486. − 110. i)12-s − 559.·13-s + (187. + 128. i)14-s + (323. + 217. i)15-s + (−757. + 689. i)16-s − 1.56e3i·17-s + ⋯
L(s)  = 1  + (−0.565 + 0.824i)2-s + (0.558 − 0.829i)3-s + (−0.360 − 0.932i)4-s + 0.447i·5-s + (0.368 + 0.929i)6-s − 0.310i·7-s + (0.973 + 0.229i)8-s + (−0.376 − 0.926i)9-s + (−0.368 − 0.252i)10-s − 1.00·11-s + (−0.975 − 0.221i)12-s − 0.918·13-s + (0.255 + 0.175i)14-s + (0.371 + 0.249i)15-s + (−0.739 + 0.673i)16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.221 + 0.975i$
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: \((2,\ 60,\ (\ :5/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.515549 - 0.645573i\)
\(L(\frac12)\) \(\approx\) \(0.515549 - 0.645573i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.19 - 4.66i)T \)
3 \( 1 + (-8.70 + 12.9i)T \)
5 \( 1 - 25iT \)
good7 \( 1 + 40.1iT - 1.68e4T^{2} \)
11 \( 1 + 401.T + 1.61e5T^{2} \)
13 \( 1 + 559.T + 3.71e5T^{2} \)
17 \( 1 + 1.56e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.49e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.05e3T + 6.43e6T^{2} \)
29 \( 1 - 4.04e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.99e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.04e4T + 6.93e7T^{2} \)
41 \( 1 - 1.25e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.07e4iT - 1.47e8T^{2} \)
47 \( 1 + 8.34e3T + 2.29e8T^{2} \)
53 \( 1 - 7.97e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.62e4T + 7.14e8T^{2} \)
61 \( 1 - 4.83e4T + 8.44e8T^{2} \)
67 \( 1 - 3.07e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.75e4T + 1.80e9T^{2} \)
73 \( 1 + 4.77e4T + 2.07e9T^{2} \)
79 \( 1 - 2.75e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.89e4T + 3.93e9T^{2} \)
89 \( 1 + 1.05e5iT - 5.58e9T^{2} \)
97 \( 1 - 9.95e4T + 8.58e9T^{2} \)
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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

−13.90314695783060025779585262965, −13.07007053582088677691683485617, −11.36839837761825992952108625138, −9.958436706946573766910175411768, −8.805930847182156097117021840379, −7.44089335962420744940930086641, −6.92918283542034648282622291479, −5.13823516309470347643265080448, −2.56809511831127383179104281460, −0.42226696860418336641241415677, 2.13673844520435683206233993090, 3.69848877477374733089232117396, 5.17210244086294246911774390963, 7.84800654485915168613498037449, 8.659632365362367530305984242289, 9.929921481628242712391370090849, 10.58551510240783972537895649162, 12.11211499197906331396097755468, 13.05535399530824553885186483071, 14.38181575189585352692610646362

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