↑

Properties

Label 2-270-135.122-c1-0-2
Degree $2$
Conductor $270$
Sign $-0.0544 - 0.998i$
Analytic cond. $2.15596$
Root an. cond. $1.46831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.819 + 0.573i)2-s + (−1.38 − 1.04i)3-s + (0.342 + 0.939i)4-s + (−2.17 + 0.533i)5-s + (−0.537 − 1.64i)6-s + (1.71 + 3.68i)7-s + (−0.258 + 0.965i)8-s + (0.835 + 2.88i)9-s + (−2.08 − 0.808i)10-s + (−0.396 − 0.472i)11-s + (0.503 − 1.65i)12-s + (2.48 + 3.55i)13-s + (−0.705 + 3.99i)14-s + (3.56 + 1.51i)15-s + (−0.766 + 0.642i)16-s + (−0.235 − 0.880i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (−0.799 − 0.600i)3-s + (0.171 + 0.469i)4-s + (−0.971 + 0.238i)5-s + (−0.219 − 0.672i)6-s + (0.648 + 1.39i)7-s + (−0.0915 + 0.341i)8-s + (0.278 + 0.960i)9-s + (−0.659 − 0.255i)10-s + (−0.119 − 0.142i)11-s + (0.145 − 0.478i)12-s + (0.690 + 0.985i)13-s + (−0.188 + 1.06i)14-s + (0.919 + 0.392i)15-s + (−0.191 + 0.160i)16-s + (−0.0572 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0544 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0544 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $-0.0544 - 0.998i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ -0.0544 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763555 + 0.806344i\)
\(L(\frac12)\) \(\approx\) \(0.763555 + 0.806344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (1.38 + 1.04i)T \)
5 \( 1 + (2.17 - 0.533i)T \)
good7 \( 1 + (-1.71 - 3.68i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (0.396 + 0.472i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.48 - 3.55i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.235 + 0.880i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.18 - 2.99i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.35 - 2.03i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.993 + 5.63i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.86 - 2.49i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-2.63 + 0.705i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.18 - 0.385i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.847 + 9.68i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-0.383 + 0.179i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-5.83 - 5.83i)T + 53iT^{2} \)
59 \( 1 + (6.49 + 5.44i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-11.5 - 4.19i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.78 + 1.95i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (0.354 + 0.204i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.20 - 1.93i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.9 + 2.11i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.59 - 12.2i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-0.995 - 1.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-18.9 - 1.65i)T + (95.5 + 16.8i)T^{2} \)
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

−12.14622943311350323459627382926, −11.43734060067562678879918732241, −10.88746824914272320624694365784, −8.908757693312664863155393517134, −8.119974020612814391315826052054, −7.09619564252829703885251406228, −6.11353743213387724124727091041, −5.19852279422014784010370975524, −4.01375285038978151395242551328, −2.17833755231830496054296532561, 0.826862349641118729643588498764, 3.52661018271335369853028049646, 4.35072997702969595886944862522, 5.14530522209954189800664699396, 6.59554482634563929161917332226, 7.62863905172457167842701581110, 8.872632933144731537652354276237, 10.35388421645810520885695269808, 10.97658814054765420401899386382, 11.31051630455853425614065835961

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