Properties

Label 2-273-91.76-c1-0-4
Degree $2$
Conductor $273$
Sign $0.886 + 0.462i$
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  + (0.385 − 1.43i)2-s + (−0.866 + 0.5i)3-s + (−0.189 − 0.109i)4-s + (−1.07 + 1.07i)5-s + (0.385 + 1.43i)6-s + (1.81 + 1.92i)7-s + (1.87 − 1.87i)8-s + (0.499 − 0.866i)9-s + (1.12 + 1.95i)10-s + (1.68 + 0.451i)11-s + 0.218·12-s + (3.51 + 0.818i)13-s + (3.46 − 1.87i)14-s + (0.392 − 1.46i)15-s + (−2.19 − 3.80i)16-s + (1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.272 − 1.01i)2-s + (−0.499 + 0.288i)3-s + (−0.0945 − 0.0545i)4-s + (−0.479 + 0.479i)5-s + (0.157 + 0.587i)6-s + (0.686 + 0.726i)7-s + (0.663 − 0.663i)8-s + (0.166 − 0.288i)9-s + (0.357 + 0.618i)10-s + (0.508 + 0.136i)11-s + 0.0630·12-s + (0.973 + 0.227i)13-s + (0.926 − 0.500i)14-s + (0.101 − 0.378i)15-s + (−0.548 − 0.950i)16-s + (0.348 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.886 + 0.462i$
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: \((2,\ 273,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39565 - 0.341886i\)
\(L(\frac12)\) \(\approx\) \(1.39565 - 0.341886i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 - 1.92i)T \)
13 \( 1 + (-3.51 - 0.818i)T \)
good2 \( 1 + (-0.385 + 1.43i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.07 - 1.07i)T - 5iT^{2} \)
11 \( 1 + (-1.68 - 0.451i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.389 - 1.45i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.21 - 1.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.65 + 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.32 + 1.32i)T - 31iT^{2} \)
37 \( 1 + (5.83 + 1.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.10 + 0.830i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.29 - 1.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.86 + 5.86i)T + 47iT^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 + (-0.967 + 0.259i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.0305 + 0.0176i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 - 4.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.4 + 3.05i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.3 + 11.3i)T + 73iT^{2} \)
79 \( 1 + 3.53T + 79T^{2} \)
83 \( 1 + (10.8 - 10.8i)T - 83iT^{2} \)
89 \( 1 + (-2.02 + 7.54i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.46 + 9.21i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.61888027826307767089872497255, −11.27331863565717337925132183484, −10.28627809739314068253397341628, −9.257892513543086709155332638266, −7.959367696275034295675927712557, −6.84462148628031353102506831533, −5.58077747361593830187392204193, −4.24406831317362405070756718486, −3.29080518190934372216717423460, −1.70048581179412321598874658827, 1.39968576577045082891098353025, 3.99551993020477267918405200050, 4.99501311272417231661825897399, 6.07519031898602388789362371203, 6.94272482882089836177849923154, 7.969415820306912400006502692445, 8.557393561305785475901864958197, 10.34850030000628007433886073812, 11.09983813862755301284538815362, 11.93117358432627927489303332823

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