↑

Properties

Label 2-273-91.19-c1-0-2
Degree $2$
Conductor $273$
Sign $0.673 - 0.738i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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  + (−1.53 − 0.409i)2-s − i·3-s + (0.440 + 0.254i)4-s + (−3.63 + 0.973i)5-s + (−0.409 + 1.53i)6-s + (−1.31 − 2.29i)7-s + (1.66 + 1.66i)8-s − 9-s + 5.95·10-s + (2.38 + 2.38i)11-s + (0.254 − 0.440i)12-s + (3.03 + 1.94i)13-s + (1.06 + 4.05i)14-s + (0.973 + 3.63i)15-s + (−2.37 − 4.12i)16-s + (−1.66 + 2.89i)17-s + ⋯
L(s)  = 1  + (−1.08 − 0.289i)2-s − 0.577i·3-s + (0.220 + 0.127i)4-s + (−1.62 + 0.435i)5-s + (−0.167 + 0.624i)6-s + (−0.495 − 0.868i)7-s + (0.590 + 0.590i)8-s − 0.333·9-s + 1.88·10-s + (0.718 + 0.718i)11-s + (0.0734 − 0.127i)12-s + (0.842 + 0.538i)13-s + (0.284 + 1.08i)14-s + (0.251 + 0.938i)15-s + (−0.594 − 1.03i)16-s + (−0.404 + 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.673 - 0.738i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.673 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308924 + 0.136381i\)
\(L(\frac12)\) \(\approx\) \(0.308924 + 0.136381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (1.31 + 2.29i)T \)
13 \( 1 + (-3.03 - 1.94i)T \)
good2 \( 1 + (1.53 + 0.409i)T + (1.73 + i)T^{2} \)
5 \( 1 + (3.63 - 0.973i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.38 - 2.38i)T + 11iT^{2} \)
17 \( 1 + (1.66 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.537 - 0.537i)T + 19iT^{2} \)
23 \( 1 + (-3.08 + 1.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.42 - 2.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.38 - 8.90i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.93 - 10.9i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.84 - 1.83i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.9 + 6.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.356 - 1.32i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.59 + 6.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.816 - 3.04i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 1.04iT - 61T^{2} \)
67 \( 1 + (2.09 - 2.09i)T - 67iT^{2} \)
71 \( 1 + (7.00 + 1.87i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.08 - 0.559i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.54 - 9.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.51 - 1.51i)T + 83iT^{2} \)
89 \( 1 + (3.05 + 0.819i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.23 + 12.0i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.75296762677755645527079551954, −10.98270091913392027410986371661, −10.29791032229151092454244337958, −8.955377817781037839757066708547, −8.279946827755491045146963891181, −7.19699794608416569745469604749, −6.75024688056930113658654408566, −4.51048838128762082693988418452, −3.45334567470657282524513875339, −1.31304660718358167479177622144, 0.42922094199379750881522164146, 3.39802601287147837378254843782, 4.29149881835076356734213134237, 5.83146710645449170939540802204, 7.25964971461396850407892470713, 8.167283672444203108594807022115, 8.966349676564580254713339189044, 9.374689537894174051866955498044, 10.92003401563759587376584769397, 11.45817947631143374472700208271

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