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Properties

Label 2-273-39.5-c1-0-0
Degree $2$
Conductor $273$
Sign $0.915 - 0.402i$
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  + (−1.76 − 1.76i)2-s + (−0.743 − 1.56i)3-s + 4.23i·4-s + (−0.269 − 0.269i)5-s + (−1.44 + 4.07i)6-s + (−0.707 − 0.707i)7-s + (3.93 − 3.93i)8-s + (−1.89 + 2.32i)9-s + 0.951i·10-s + (−4.05 + 4.05i)11-s + (6.61 − 3.14i)12-s + (1.60 + 3.22i)13-s + 2.49i·14-s + (−0.221 + 0.622i)15-s − 5.43·16-s + 1.75·17-s + ⋯
L(s)  = 1  + (−1.24 − 1.24i)2-s + (−0.429 − 0.903i)3-s + 2.11i·4-s + (−0.120 − 0.120i)5-s + (−0.591 + 1.66i)6-s + (−0.267 − 0.267i)7-s + (1.39 − 1.39i)8-s + (−0.631 + 0.775i)9-s + 0.300i·10-s + (−1.22 + 1.22i)11-s + (1.91 − 0.907i)12-s + (0.445 + 0.895i)13-s + 0.667i·14-s + (−0.0571 + 0.160i)15-s − 1.35·16-s + 0.425·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.915 - 0.402i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203886 + 0.0428215i\)
\(L(\frac12)\) \(\approx\) \(0.203886 + 0.0428215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.743 + 1.56i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-1.60 - 3.22i)T \)
good2 \( 1 + (1.76 + 1.76i)T + 2iT^{2} \)
5 \( 1 + (0.269 + 0.269i)T + 5iT^{2} \)
11 \( 1 + (4.05 - 4.05i)T - 11iT^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 + (2.02 - 2.02i)T - 19iT^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 + (2.13 - 2.13i)T - 31iT^{2} \)
37 \( 1 + (2.60 + 2.60i)T + 37iT^{2} \)
41 \( 1 + (-5.68 - 5.68i)T + 41iT^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (7.30 - 7.30i)T - 47iT^{2} \)
53 \( 1 - 6.27iT - 53T^{2} \)
59 \( 1 + (7.32 - 7.32i)T - 59iT^{2} \)
61 \( 1 + 4.57T + 61T^{2} \)
67 \( 1 + (-9.02 + 9.02i)T - 67iT^{2} \)
71 \( 1 + (-0.688 - 0.688i)T + 71iT^{2} \)
73 \( 1 + (7.93 + 7.93i)T + 73iT^{2} \)
79 \( 1 + 3.01T + 79T^{2} \)
83 \( 1 + (8.09 + 8.09i)T + 83iT^{2} \)
89 \( 1 + (3.81 - 3.81i)T - 89iT^{2} \)
97 \( 1 + (7.04 - 7.04i)T - 97iT^{2} \)
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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

−11.90023045533582719944079716777, −10.93167032299020172613249571851, −10.23534640860446496913254266903, −9.305528174162385822317958589313, −8.043870832101199449624619731336, −7.59074696941278477676182389520, −6.27198356347627719300293588601, −4.46009680729594113397764279600, −2.69815704764163005436377764673, −1.57695398108746821321792168215, 0.25104200175606929765447012594, 3.32555759365368357671959938550, 5.39341618363209756111118073714, 5.72065316050783805787667333211, 6.98796559957116983659034920137, 8.225676485270996847553966012909, 8.758236451265926157722108011641, 9.853792970806347908791581632101, 10.59566623896126209454594582219, 11.22250542084184502283149856687

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