↑

Properties

Label 2-270-15.14-c2-0-11
Degree $2$
Conductor $270$
Sign $0.882 + 0.469i$
Analytic cond. $7.35696$
Root an. cond. $2.71237$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + (4.41 + 2.34i)5-s − 13.6i·7-s + 2.82·8-s + (6.24 + 3.32i)10-s − 12.3i·11-s + 17.0i·13-s − 19.3i·14-s + 4.00·16-s + 6.89·17-s − 7.24·19-s + (8.82 + 4.69i)20-s − 17.4i·22-s + 34.7·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s + (0.882 + 0.469i)5-s − 1.95i·7-s + 0.353·8-s + (0.624 + 0.332i)10-s − 1.11i·11-s + 1.30i·13-s − 1.38i·14-s + 0.250·16-s + 0.405·17-s − 0.381·19-s + (0.441 + 0.234i)20-s − 0.791i·22-s + 1.51·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.882 + 0.469i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.882 + 0.469i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.71312 - 0.676779i\)
\(L(\frac12)\) \(\approx\) \(2.71312 - 0.676779i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (-4.41 - 2.34i)T \)
good7 \( 1 + 13.6iT - 49T^{2} \)
11 \( 1 + 12.3iT - 121T^{2} \)
13 \( 1 - 17.0iT - 169T^{2} \)
17 \( 1 - 6.89T + 289T^{2} \)
19 \( 1 + 7.24T + 361T^{2} \)
23 \( 1 - 34.7T + 529T^{2} \)
29 \( 1 - 21.1iT - 841T^{2} \)
31 \( 1 + 38.2T + 961T^{2} \)
37 \( 1 - 21.5iT - 1.36e3T^{2} \)
41 \( 1 + 36.3iT - 1.68e3T^{2} \)
43 \( 1 + 6.23iT - 1.84e3T^{2} \)
47 \( 1 + 40.2T + 2.20e3T^{2} \)
53 \( 1 - 38.2T + 2.80e3T^{2} \)
59 \( 1 - 41.6iT - 3.48e3T^{2} \)
61 \( 1 + 15.0T + 3.72e3T^{2} \)
67 \( 1 - 128. iT - 4.48e3T^{2} \)
71 \( 1 - 104. iT - 5.04e3T^{2} \)
73 \( 1 + 2.11iT - 5.32e3T^{2} \)
79 \( 1 + 44.0T + 6.24e3T^{2} \)
83 \( 1 + 55.0T + 6.88e3T^{2} \)
89 \( 1 + 68.1iT - 7.92e3T^{2} \)
97 \( 1 + 101. iT - 9.40e3T^{2} \)
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

−11.36118465453758383579685178465, −10.84354365827323052476974429431, −10.02046790530302211228153409666, −8.800392186624120245410938396897, −7.17260432168819703158296280509, −6.78804375527854317907766419803, −5.51090475078865156290186906006, −4.21093837532320859645935550454, −3.18422179962100390402530256259, −1.36308158674119424167385501271, 1.93732026409224515888187651683, 2.95677360655595698093305842488, 4.94701032449255639708063149594, 5.48588888482409586482675146182, 6.41924113980946893823344177050, 7.907132758885441625925730057259, 9.032688784319610829168448445392, 9.762561430130505667733978252728, 10.96082217037739127428287567064, 12.19785072131953477777015872910

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