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Properties

Label 2-273-21.20-c1-0-16
Degree $2$
Conductor $273$
Sign $0.391 - 0.920i$
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  + 0.633i·2-s + (0.915 + 1.47i)3-s + 1.59·4-s − 0.119·5-s + (−0.931 + 0.579i)6-s + (2.16 − 1.52i)7-s + 2.28i·8-s + (−1.32 + 2.69i)9-s − 0.0758i·10-s − 4.57i·11-s + (1.46 + 2.35i)12-s + i·13-s + (0.963 + 1.37i)14-s + (−0.109 − 0.176i)15-s + 1.75·16-s − 6.89·17-s + ⋯
L(s)  = 1  + 0.448i·2-s + (0.528 + 0.848i)3-s + 0.799·4-s − 0.0535·5-s + (−0.380 + 0.236i)6-s + (0.818 − 0.574i)7-s + 0.806i·8-s + (−0.441 + 0.897i)9-s − 0.0239i·10-s − 1.37i·11-s + (0.422 + 0.678i)12-s + 0.277i·13-s + (0.257 + 0.366i)14-s + (−0.0282 − 0.0454i)15-s + 0.438·16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49989 + 0.992382i\)
\(L(\frac12)\) \(\approx\) \(1.49989 + 0.992382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.915 - 1.47i)T \)
7 \( 1 + (-2.16 + 1.52i)T \)
13 \( 1 - iT \)
good2 \( 1 - 0.633iT - 2T^{2} \)
5 \( 1 + 0.119T + 5T^{2} \)
11 \( 1 + 4.57iT - 11T^{2} \)
17 \( 1 + 6.89T + 17T^{2} \)
19 \( 1 + 0.125iT - 19T^{2} \)
23 \( 1 + 1.84iT - 23T^{2} \)
29 \( 1 - 3.43iT - 29T^{2} \)
31 \( 1 - 0.162iT - 31T^{2} \)
37 \( 1 + 6.26T + 37T^{2} \)
41 \( 1 - 7.36T + 41T^{2} \)
43 \( 1 - 1.26T + 43T^{2} \)
47 \( 1 - 2.57T + 47T^{2} \)
53 \( 1 + 6.10iT - 53T^{2} \)
59 \( 1 - 7.02T + 59T^{2} \)
61 \( 1 + 2.09iT - 61T^{2} \)
67 \( 1 + 1.47T + 67T^{2} \)
71 \( 1 - 5.86iT - 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 + 9.95T + 79T^{2} \)
83 \( 1 + 7.28T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 - 5.75iT - 97T^{2} \)
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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

−11.63220579771041549034712636146, −11.04870034953599397901934364421, −10.44556595891066514669922573611, −8.928703329687083798220663138077, −8.295900043451645888190201067307, −7.29152928278040970509782505873, −6.08134116504600933731623986934, −4.89615796353159851430368182674, −3.67830661549334083895365142781, −2.21995157310464638029036344154, 1.79846108328872857136510634012, 2.51056747074735903649792529173, 4.21241803312872022536761339927, 5.86852180667357240883171548084, 7.00885151299314774954942357267, 7.68734194398405122073836186361, 8.800347581247185671933473667136, 9.838631226960339356664502297450, 11.07923055734925199763788103098, 11.81291188057922810441884911233

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