↑

Properties

Label 2-273-273.152-c1-0-19
Degree $2$
Conductor $273$
Sign $0.924 + 0.381i$
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  + (−0.667 + 0.385i)2-s + (1.00 − 1.40i)3-s + (−0.702 + 1.21i)4-s + (0.907 − 1.57i)5-s + (−0.130 + 1.32i)6-s + (1.94 + 1.78i)7-s − 2.62i·8-s + (−0.964 − 2.84i)9-s + 1.39i·10-s − 0.582i·11-s + (1.00 + 2.21i)12-s + (3.53 + 0.728i)13-s + (−1.99 − 0.442i)14-s + (−1.29 − 2.86i)15-s + (−0.392 − 0.680i)16-s + (−0.479 + 0.829i)17-s + ⋯
L(s)  = 1  + (−0.472 + 0.272i)2-s + (0.582 − 0.812i)3-s + (−0.351 + 0.608i)4-s + (0.405 − 0.703i)5-s + (−0.0534 + 0.542i)6-s + (0.736 + 0.676i)7-s − 0.928i·8-s + (−0.321 − 0.946i)9-s + 0.442i·10-s − 0.175i·11-s + (0.289 + 0.640i)12-s + (0.979 + 0.201i)13-s + (−0.532 − 0.118i)14-s + (−0.335 − 0.739i)15-s + (−0.0982 − 0.170i)16-s + (−0.116 + 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.924 + 0.381i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.924 + 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23893 - 0.245804i\)
\(L(\frac12)\) \(\approx\) \(1.23893 - 0.245804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.00 + 1.40i)T \)
7 \( 1 + (-1.94 - 1.78i)T \)
13 \( 1 + (-3.53 - 0.728i)T \)
good2 \( 1 + (0.667 - 0.385i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.907 + 1.57i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.582iT - 11T^{2} \)
17 \( 1 + (0.479 - 0.829i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.92iT - 19T^{2} \)
23 \( 1 + (-7.56 + 4.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.26 + 2.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.83 - 1.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.899 - 1.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.73 - 2.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.367 + 0.636i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.49 - 6.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.08 - 2.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.82 - 6.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 3.82iT - 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (9.73 - 5.61i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.30 - 4.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.35 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.45T + 83T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.89 + 4.55i)T + (48.5 - 84.0i)T^{2} \)
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

−12.04334004527806081916628514999, −11.03398019559357400820056911598, −9.254935268324535892143078858510, −8.879832234993674837023570151229, −8.208396182165102195658060637800, −7.18687780586697514311254323271, −6.02057761827621362786964614838, −4.60973647788387696773095062365, −3.04760545010185883509705610455, −1.35852628583970183163368662436, 1.73803409946488639742259150939, 3.37145380624365688802916653994, 4.70638243720170058813576881453, 5.72152385873393635558762176657, 7.27841047933777935161175039345, 8.414886900834145846961739005827, 9.241545126563646949685915696871, 10.13242778402532643362276544386, 10.85770069933285304219087981356, 11.26728406463004315963429472070

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