↑

Properties

Label 2-60-60.59-c3-0-14
Degree $2$
Conductor $60$
Sign $0.955 + 0.295i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43 + 2.43i)2-s + (−5.17 + 0.459i)3-s + (−3.85 − 7.00i)4-s + (8.70 − 7.01i)5-s + (6.32 − 13.2i)6-s − 7.71·7-s + (22.6 + 0.684i)8-s + (26.5 − 4.75i)9-s + (4.57 + 31.2i)10-s + 38.0·11-s + (23.1 + 34.4i)12-s − 63.3i·13-s + (11.1 − 18.7i)14-s + (−41.8 + 40.3i)15-s + (−34.2 + 54.0i)16-s + 10.2·17-s + ⋯
L(s)  = 1  + (−0.508 + 0.860i)2-s + (−0.996 + 0.0883i)3-s + (−0.482 − 0.875i)4-s + (0.778 − 0.627i)5-s + (0.430 − 0.902i)6-s − 0.416·7-s + (0.999 + 0.0302i)8-s + (0.984 − 0.176i)9-s + (0.144 + 0.989i)10-s + 1.04·11-s + (0.557 + 0.829i)12-s − 1.35i·13-s + (0.211 − 0.358i)14-s + (−0.719 + 0.694i)15-s + (−0.534 + 0.845i)16-s + 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.955 + 0.295i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.809137 - 0.122334i\)
\(L(\frac12)\) \(\approx\) \(0.809137 - 0.122334i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.43 - 2.43i)T \)
3 \( 1 + (5.17 - 0.459i)T \)
5 \( 1 + (-8.70 + 7.01i)T \)
good7 \( 1 + 7.71T + 343T^{2} \)
11 \( 1 - 38.0T + 1.33e3T^{2} \)
13 \( 1 + 63.3iT - 2.19e3T^{2} \)
17 \( 1 - 10.2T + 4.91e3T^{2} \)
19 \( 1 + 99.5iT - 6.85e3T^{2} \)
23 \( 1 + 133. iT - 1.21e4T^{2} \)
29 \( 1 - 197. iT - 2.43e4T^{2} \)
31 \( 1 + 13.9iT - 2.97e4T^{2} \)
37 \( 1 - 272. iT - 5.06e4T^{2} \)
41 \( 1 + 166. iT - 6.89e4T^{2} \)
43 \( 1 - 273.T + 7.95e4T^{2} \)
47 \( 1 - 69.1iT - 1.03e5T^{2} \)
53 \( 1 - 300.T + 1.48e5T^{2} \)
59 \( 1 - 618.T + 2.05e5T^{2} \)
61 \( 1 + 439.T + 2.26e5T^{2} \)
67 \( 1 - 23.6T + 3.00e5T^{2} \)
71 \( 1 + 827.T + 3.57e5T^{2} \)
73 \( 1 + 152. iT - 3.89e5T^{2} \)
79 \( 1 + 421. iT - 4.93e5T^{2} \)
83 \( 1 - 709. iT - 5.71e5T^{2} \)
89 \( 1 - 1.43e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.08e3iT - 9.12e5T^{2} \)
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

−14.80196964393174498038183223257, −13.39940638434020140117987600127, −12.44909684229341057714260977205, −10.76245698080709669246500118742, −9.796746236549171340523389134645, −8.717990162120686121890981313807, −6.91621750368591260942360761438, −5.91546775555718760589537021141, −4.78496818813865928032982539707, −0.852767242424988624848158806238, 1.69176601896376072311430703681, 3.97241607699124734107753323424, 6.02401533015839694363195775447, 7.24402389947492333854552105985, 9.328348033850604876056523306699, 10.04474660029562827248857053773, 11.29532956116003008027124972872, 12.01564836604010294814908227894, 13.28380085783695257365501801412, 14.33059319538243816423857771829

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