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Properties

Label 2-273-273.185-c1-0-13
Degree $2$
Conductor $273$
Sign $-0.998 - 0.0491i$
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  + (−2.25 − 1.30i)2-s + (−1.59 − 0.680i)3-s + (2.38 + 4.12i)4-s + (1.54 + 2.67i)5-s + (2.70 + 3.60i)6-s + (−2.63 + 0.251i)7-s − 7.19i·8-s + (2.07 + 2.16i)9-s − 8.04i·10-s − 3.27i·11-s + (−0.984 − 8.19i)12-s + (−3.57 − 0.487i)13-s + (6.25 + 2.85i)14-s + (−0.639 − 5.31i)15-s + (−4.58 + 7.94i)16-s + (−1.92 − 3.32i)17-s + ⋯
L(s)  = 1  + (−1.59 − 0.919i)2-s + (−0.919 − 0.393i)3-s + (1.19 + 2.06i)4-s + (0.691 + 1.19i)5-s + (1.10 + 1.47i)6-s + (−0.995 + 0.0951i)7-s − 2.54i·8-s + (0.691 + 0.722i)9-s − 2.54i·10-s − 0.986i·11-s + (−0.284 − 2.36i)12-s + (−0.990 − 0.135i)13-s + (1.67 + 0.763i)14-s + (−0.165 − 1.37i)15-s + (−1.14 + 1.98i)16-s + (−0.466 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.998 - 0.0491i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.998 - 0.0491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00328842 + 0.133617i\)
\(L(\frac12)\) \(\approx\) \(0.00328842 + 0.133617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 + 0.680i)T \)
7 \( 1 + (2.63 - 0.251i)T \)
13 \( 1 + (3.57 + 0.487i)T \)
good2 \( 1 + (2.25 + 1.30i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.54 - 2.67i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.27iT - 11T^{2} \)
17 \( 1 + (1.92 + 3.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.49iT - 19T^{2} \)
23 \( 1 + (-2.95 - 1.70i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.85 - 2.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.89 + 3.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.981 - 1.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.81 + 8.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.70 + 6.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.63 + 2.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.67 + 2.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.19 + 2.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.27iT - 61T^{2} \)
67 \( 1 + 9.76T + 67T^{2} \)
71 \( 1 + (3.88 + 2.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.06 - 1.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.24 + 5.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.84T + 83T^{2} \)
89 \( 1 + (3.58 - 6.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.4 + 6.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.10504912234286763340627649608, −10.53351945043352509392894849374, −9.749132434090954763436893488328, −8.946508624664546897960168277920, −7.24976429595481617834946268488, −6.97616798718371331442863367609, −5.69624419165769570847444702517, −3.21317636113165595395472917136, −2.22784931715128668479606778588, −0.18488255586017797297388412360, 1.58082867438451729778531419730, 4.66430892420752324524893963123, 5.70534290931748067171379080570, 6.53764727136085435012188703126, 7.46296601255994147790427599820, 8.849946655356658178912278455042, 9.622621309921187646799741171834, 9.923873268378036994225081729341, 10.96326034070058022459050718838, 12.40990237197575516872136568506

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