↑

Properties

Label 2-60-20.19-c4-0-14
Degree $2$
Conductor $60$
Sign $0.720 - 0.693i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.90 + 0.854i)2-s + 5.19·3-s + (14.5 + 6.67i)4-s + (−9.11 + 23.2i)5-s + (20.3 + 4.44i)6-s + 10.5·7-s + (51.1 + 38.5i)8-s + 27·9-s + (−55.5 + 83.1i)10-s + 38.4i·11-s + (75.5 + 34.7i)12-s − 273. i·13-s + (41.1 + 8.99i)14-s + (−47.3 + 120. i)15-s + (166. + 194. i)16-s − 256. i·17-s + ⋯
L(s)  = 1  + (0.976 + 0.213i)2-s + 0.577·3-s + (0.908 + 0.417i)4-s + (−0.364 + 0.931i)5-s + (0.564 + 0.123i)6-s + 0.214·7-s + (0.798 + 0.601i)8-s + 0.333·9-s + (−0.555 + 0.831i)10-s + 0.317i·11-s + (0.524 + 0.241i)12-s − 1.61i·13-s + (0.209 + 0.0458i)14-s + (−0.210 + 0.537i)15-s + (0.651 + 0.758i)16-s − 0.887i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.95242 + 1.19110i\)
\(L(\frac12)\) \(\approx\) \(2.95242 + 1.19110i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.90 - 0.854i)T \)
3 \( 1 - 5.19T \)
5 \( 1 + (9.11 - 23.2i)T \)
good7 \( 1 - 10.5T + 2.40e3T^{2} \)
11 \( 1 - 38.4iT - 1.46e4T^{2} \)
13 \( 1 + 273. iT - 2.85e4T^{2} \)
17 \( 1 + 256. iT - 8.35e4T^{2} \)
19 \( 1 - 257. iT - 1.30e5T^{2} \)
23 \( 1 + 168.T + 2.79e5T^{2} \)
29 \( 1 + 684.T + 7.07e5T^{2} \)
31 \( 1 + 754. iT - 9.23e5T^{2} \)
37 \( 1 + 1.65e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.20e3T + 2.82e6T^{2} \)
43 \( 1 + 3.41e3T + 3.41e6T^{2} \)
47 \( 1 - 3.24e3T + 4.87e6T^{2} \)
53 \( 1 - 639. iT - 7.89e6T^{2} \)
59 \( 1 - 6.77e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.88e3T + 1.38e7T^{2} \)
67 \( 1 - 6.75e3T + 2.01e7T^{2} \)
71 \( 1 - 5.87e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.21e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.38e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.47e3T + 4.74e7T^{2} \)
89 \( 1 + 4.60e3T + 6.27e7T^{2} \)
97 \( 1 - 758. iT - 8.85e7T^{2} \)
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

−14.59627241976175370663638013505, −13.53379434050658124389140135672, −12.42634364204018557226916848603, −11.21341645910940785293487088950, −10.09494326304649639626299514527, −8.022715390258785976601602705367, −7.21685247092542909245738377889, −5.62740078986747482760366797514, −3.84657676333829598874258292482, −2.58813476472120665807580994115, 1.73282554862799239481712637241, 3.78014843628686995872021887813, 4.92268600397318637282434914554, 6.66447380758632541949928935711, 8.203692005874138633727088173491, 9.477009398642780574403992632807, 11.13756765158672332049845855393, 12.11700938798464776838929925691, 13.18371999601605009882043154354, 14.05108575379615622741437057874

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