↑

Properties

Label 2-60-12.11-c5-0-12
Degree $2$
Conductor $60$
Sign $-0.618 - 0.785i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.53 + 3.38i)2-s + (12.7 + 9.00i)3-s + (9.09 − 30.6i)4-s + 25i·5-s + (−88.1 + 2.25i)6-s + 20.4i·7-s + (62.6 + 169. i)8-s + (80.8 + 229. i)9-s + (−84.6 − 113. i)10-s + 681.·11-s + (391. − 308. i)12-s − 767.·13-s + (−69.3 − 92.8i)14-s + (−225. + 318. i)15-s + (−858. − 558. i)16-s + 2.03e3i·17-s + ⋯
L(s)  = 1  + (−0.801 + 0.598i)2-s + (0.816 + 0.577i)3-s + (0.284 − 0.958i)4-s + 0.447i·5-s + (−0.999 + 0.0255i)6-s + 0.158i·7-s + (0.345 + 0.938i)8-s + (0.332 + 0.942i)9-s + (−0.267 − 0.358i)10-s + 1.69·11-s + (0.785 − 0.618i)12-s − 1.25·13-s + (−0.0945 − 0.126i)14-s + (−0.258 + 0.365i)15-s + (−0.838 − 0.544i)16-s + 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.618 - 0.785i$
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: \((2,\ 60,\ (\ :5/2),\ -0.618 - 0.785i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.620680 + 1.27846i\)
\(L(\frac12)\) \(\approx\) \(0.620680 + 1.27846i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.53 - 3.38i)T \)
3 \( 1 + (-12.7 - 9.00i)T \)
5 \( 1 - 25iT \)
good7 \( 1 - 20.4iT - 1.68e4T^{2} \)
11 \( 1 - 681.T + 1.61e5T^{2} \)
13 \( 1 + 767.T + 3.71e5T^{2} \)
17 \( 1 - 2.03e3iT - 1.41e6T^{2} \)
19 \( 1 - 321. iT - 2.47e6T^{2} \)
23 \( 1 + 3.92e3T + 6.43e6T^{2} \)
29 \( 1 + 1.21e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.33e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.11e3T + 6.93e7T^{2} \)
41 \( 1 - 442. iT - 1.15e8T^{2} \)
43 \( 1 + 1.10e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.95e4T + 2.29e8T^{2} \)
53 \( 1 - 1.63e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.49e3T + 7.14e8T^{2} \)
61 \( 1 - 1.44e4T + 8.44e8T^{2} \)
67 \( 1 + 2.36e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.00e4T + 1.80e9T^{2} \)
73 \( 1 + 7.20e3T + 2.07e9T^{2} \)
79 \( 1 + 5.16e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.19e4T + 3.93e9T^{2} \)
89 \( 1 + 9.77e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.06e5T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.66641324673062918998700105743, −14.04242743875118765378312630786, −12.05517759319813723404676200605, −10.54139368058800417695049658885, −9.675401380259145618160314763482, −8.651773280103614972351588375841, −7.47860803062445050703169414354, −6.08167942079113610078851462879, −4.10825963426876769416195015932, −1.99135711877463691211356931075, 0.824243023818858160689781840204, 2.38905404435278533909118486219, 4.07858131339407384266931642191, 6.81584768393656216666956833090, 7.82671123814967569626208134702, 9.184028967662789004139290790569, 9.698630481473697757330800207021, 11.71485254078633941744839570905, 12.24901822452483007632415930811, 13.59900636852493599194203179205

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