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Properties

Label 2-273-39.8-c1-0-27
Degree $2$
Conductor $273$
Sign $-0.890 + 0.455i$
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 + (−4.07 − 1.44i)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 + (−0.622 − 0.221i)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 + (−1.66 − 0.591i)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 + (−0.160 − 0.0571i)15-s − 1.35·16-s − 0.425·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486029 - 2.01601i\)
\(L(\frac12)\) \(\approx\) \(0.486029 - 2.01601i\)
\(L(\frac{3}{2})\) 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

−11.77139110364617191316253827712, −11.01424080058455338889962779249, −10.01362830118911225538235062900, −8.861369025535524608917387020439, −7.18621167526012750678895103394, −6.17197837363982575250943142054, −5.23068563404101039887661630263, −4.05922651008139599345074012921, −2.59431458382576728399694735425, −1.39719232788617096260206568997, 3.49261365029809051605831702986, 4.09172365259762723087477582571, 5.27033710678474476549403073278, 6.36575786735264478249423952521, 6.69332636380483898218988682910, 8.427544495007666277241103155643, 9.155144064400325440494132982372, 10.62805072282807314813050012152, 11.56497414649759894570891849700, 12.39730766294775497005285359440

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