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Properties

Label 2-273-273.68-c1-0-7
Degree $2$
Conductor $273$
Sign $-0.923 + 0.383i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.39i·2-s + (1.36 + 1.05i)3-s − 3.74·4-s + (−1.10 + 1.90i)5-s + (−2.54 + 3.28i)6-s + (−1.62 − 2.08i)7-s − 4.18i·8-s + (0.753 + 2.90i)9-s + (−4.56 − 2.63i)10-s + (−1.30 − 0.751i)11-s + (−5.12 − 3.96i)12-s + (3.47 + 0.976i)13-s + (5.00 − 3.89i)14-s + (−3.52 + 1.44i)15-s + 2.53·16-s + 4.90·17-s + ⋯
L(s)  = 1  + 1.69i·2-s + (0.790 + 0.611i)3-s − 1.87·4-s + (−0.492 + 0.852i)5-s + (−1.03 + 1.34i)6-s + (−0.614 − 0.789i)7-s − 1.47i·8-s + (0.251 + 0.967i)9-s + (−1.44 − 0.834i)10-s + (−0.392 − 0.226i)11-s + (−1.48 − 1.14i)12-s + (0.962 + 0.270i)13-s + (1.33 − 1.04i)14-s + (−0.911 + 0.373i)15-s + 0.632·16-s + 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.243600 - 1.22262i\)
\(L(\frac12)\) \(\approx\) \(0.243600 - 1.22262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 - 1.05i)T \)
7 \( 1 + (1.62 + 2.08i)T \)
13 \( 1 + (-3.47 - 0.976i)T \)
good2 \( 1 - 2.39iT - 2T^{2} \)
5 \( 1 + (1.10 - 1.90i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.30 + 0.751i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 + (2.77 - 1.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.94iT - 23T^{2} \)
29 \( 1 + (-0.775 + 0.447i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.33 + 1.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + (4.98 + 8.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.820 - 1.42i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.165 + 0.286i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.36 + 3.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + (-2.44 + 1.41i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 6.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.9 - 8.04i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.87 - 1.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.02 - 5.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.74T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (11.5 + 6.67i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.01045240391389085344031695558, −11.19842701585281837926262643637, −10.24596320872803532444386968791, −9.350983425714773036391232029497, −8.191705942669388643649118046933, −7.61981895942119296560539456838, −6.71102387389916001973027582482, −5.61365194965496646106683332215, −4.14726245898685262017517810848, −3.32635285131003557433472395110, 0.954450804495685414746773670056, 2.49602371616134164937760016925, 3.44083116317708112925014543708, 4.65260281724345710775741846603, 6.28633300314351924964148056333, 8.061896839011776649103127085750, 8.680285128902368095220501497581, 9.498418494272178121827722376449, 10.43752333743416989607215029392, 11.69946479922568264089304733441

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