↑

Properties

Label 2-273-91.76-c1-0-8
Degree $2$
Conductor $273$
Sign $0.989 + 0.147i$
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  + (0.0578 − 0.215i)2-s + (−0.866 + 0.5i)3-s + (1.68 + 0.975i)4-s + (1.08 − 1.08i)5-s + (0.0578 + 0.215i)6-s + (−0.725 − 2.54i)7-s + (0.624 − 0.624i)8-s + (0.499 − 0.866i)9-s + (−0.171 − 0.297i)10-s + (4.17 + 1.11i)11-s − 1.95·12-s + (−3.01 + 1.98i)13-s + (−0.591 + 0.00954i)14-s + (−0.398 + 1.48i)15-s + (1.85 + 3.20i)16-s + (3.78 − 6.55i)17-s + ⋯
L(s)  = 1  + (0.0408 − 0.152i)2-s + (−0.499 + 0.288i)3-s + (0.844 + 0.487i)4-s + (0.486 − 0.486i)5-s + (0.0236 + 0.0880i)6-s + (−0.274 − 0.961i)7-s + (0.220 − 0.220i)8-s + (0.166 − 0.288i)9-s + (−0.0543 − 0.0941i)10-s + (1.25 + 0.337i)11-s − 0.562·12-s + (−0.835 + 0.550i)13-s + (−0.157 + 0.00255i)14-s + (−0.102 + 0.383i)15-s + (0.462 + 0.801i)16-s + (0.917 − 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.989 + 0.147i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.989 + 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45529 - 0.107990i\)
\(L(\frac12)\) \(\approx\) \(1.45529 - 0.107990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.725 + 2.54i)T \)
13 \( 1 + (3.01 - 1.98i)T \)
good2 \( 1 + (-0.0578 + 0.215i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.08 + 1.08i)T - 5iT^{2} \)
11 \( 1 + (-4.17 - 1.11i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.78 + 6.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.31 - 4.91i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.85 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.66 + 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.09 + 1.09i)T - 31iT^{2} \)
37 \( 1 + (2.82 + 0.757i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.37 - 0.369i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.58 + 3.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.26 - 5.26i)T + 47iT^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + (8.73 - 2.34i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.1 + 5.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.28 - 4.78i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.25 - 1.40i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.04 - 8.04i)T + 73iT^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + (-1.13 + 1.13i)T - 83iT^{2} \)
89 \( 1 + (0.217 - 0.811i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.35 + 16.2i)T + (-84.0 + 48.5i)T^{2} \)
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

−11.96554385196086753987912475869, −11.09834550174688707252587145151, −9.822552947422091354494541298458, −9.503724035240610766623343901499, −7.68439052925882696910449998508, −6.99586986227857708749340699694, −5.92640172258634929165804203075, −4.54260731829600922732151412020, −3.42160803515362193926916079966, −1.53193350946140344500567295527, 1.72566780237312401278477804267, 3.07990985947668178752158287212, 5.16616986649935521953543938079, 6.16487797148387920434759996634, 6.57476850976317102337058388117, 7.84563966726000858568390561600, 9.209885915690839241882561620708, 10.21282366287693205635106625059, 10.96666511456079263540573968613, 12.03232138803184285768350061020

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