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Properties

Label 2-273-273.146-c1-0-17
Degree $2$
Conductor $273$
Sign $0.949 - 0.315i$
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  + (1.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (2 + 1.73i)7-s + (1.5 − 2.59i)9-s + 3.46i·12-s + (2.5 + 2.59i)13-s + (−1.99 − 3.46i)16-s + (3 + 1.73i)19-s + (4.5 + 0.866i)21-s − 5·25-s − 5.19i·27-s + (−5 + 1.73i)28-s − 8.66i·31-s + (3 + 5.19i)36-s + (−5 − 8.66i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.755 + 0.654i)7-s + (0.5 − 0.866i)9-s + 0.999i·12-s + (0.693 + 0.720i)13-s + (−0.499 − 0.866i)16-s + (0.688 + 0.397i)19-s + (0.981 + 0.188i)21-s − 25-s − 0.999i·27-s + (−0.944 + 0.327i)28-s − 1.55i·31-s + (0.5 + 0.866i)36-s + (−0.821 − 1.42i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.949 - 0.315i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.949 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59386 + 0.257748i\)
\(L(\frac12)\) \(\approx\) \(1.59386 + 0.257748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-2 - 1.73i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.5 - 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.5 - 9.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.01775081598151894906884131328, −11.37501183017857877846891693132, −9.628603214583999606269110785913, −8.936973863531753034304449028783, −8.082943810695686673275596558862, −7.45513720265543218143103192689, −6.01882596893972613950782762000, −4.44984241155225366877211282197, −3.37554555061242399061086779178, −1.94807609762442698936696248302, 1.51455789003945511277452601152, 3.42907042790280078111511858902, 4.61041336590011580804505857709, 5.48773395311062042178303329598, 7.08623592823649647770118944379, 8.253558633798219654478234338012, 8.925950451305429702959682283376, 10.16641845627850680700476578894, 10.48381003860056307863465221018, 11.64649395896150823072218025337

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