Properties

Label 2-273-39.5-c1-0-13
Degree $2$
Conductor $273$
Sign $0.173 - 0.984i$
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  + (1.08 + 1.08i)2-s + (1.14 + 1.30i)3-s + 0.353i·4-s + (0.879 + 0.879i)5-s + (−0.168 + 2.65i)6-s + (−0.707 − 0.707i)7-s + (1.78 − 1.78i)8-s + (−0.380 + 2.97i)9-s + 1.90i·10-s + (0.135 − 0.135i)11-s + (−0.460 + 0.404i)12-s + (−3.31 − 1.41i)13-s − 1.53i·14-s + (−0.136 + 2.14i)15-s + 4.58·16-s − 3.19·17-s + ⋯
L(s)  = 1  + (0.767 + 0.767i)2-s + (0.660 + 0.750i)3-s + 0.176i·4-s + (0.393 + 0.393i)5-s + (−0.0689 + 1.08i)6-s + (−0.267 − 0.267i)7-s + (0.631 − 0.631i)8-s + (−0.126 + 0.991i)9-s + 0.603i·10-s + (0.0407 − 0.0407i)11-s + (−0.132 + 0.116i)12-s + (−0.919 − 0.393i)13-s − 0.410i·14-s + (−0.0353 + 0.554i)15-s + 1.14·16-s − 0.774·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.173 - 0.984i$
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: \((2,\ 273,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74260 + 1.46297i\)
\(L(\frac12)\) \(\approx\) \(1.74260 + 1.46297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.14 - 1.30i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (3.31 + 1.41i)T \)
good2 \( 1 + (-1.08 - 1.08i)T + 2iT^{2} \)
5 \( 1 + (-0.879 - 0.879i)T + 5iT^{2} \)
11 \( 1 + (-0.135 + 0.135i)T - 11iT^{2} \)
17 \( 1 + 3.19T + 17T^{2} \)
19 \( 1 + (0.0287 - 0.0287i)T - 19iT^{2} \)
23 \( 1 + 1.05T + 23T^{2} \)
29 \( 1 + 2.50iT - 29T^{2} \)
31 \( 1 + (-0.0596 + 0.0596i)T - 31iT^{2} \)
37 \( 1 + (-1.80 - 1.80i)T + 37iT^{2} \)
41 \( 1 + (-3.83 - 3.83i)T + 41iT^{2} \)
43 \( 1 + 4.62iT - 43T^{2} \)
47 \( 1 + (-1.94 + 1.94i)T - 47iT^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 + (10.5 - 10.5i)T - 59iT^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-6.24 + 6.24i)T - 67iT^{2} \)
71 \( 1 + (7.33 + 7.33i)T + 71iT^{2} \)
73 \( 1 + (-0.460 - 0.460i)T + 73iT^{2} \)
79 \( 1 - 6.30T + 79T^{2} \)
83 \( 1 + (-7.87 - 7.87i)T + 83iT^{2} \)
89 \( 1 + (8.78 - 8.78i)T - 89iT^{2} \)
97 \( 1 + (13.0 - 13.0i)T - 97iT^{2} \)
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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

−12.42823186027879600338246312448, −10.88308644645701029440628992058, −10.13571746197256161915859595209, −9.423508225264625572216889468127, −8.060683547982639187976927692566, −7.06652832328638484871607272894, −6.01770048195857219986816192735, −4.90232575704998385764181082644, −3.98631831519070914224620214285, −2.54338181256932334877151119195, 1.85183197375237186868028817836, 2.83603848786568730960301417552, 4.13768521807400361125443962679, 5.38997236514130162441278244251, 6.76322915095215309870008922931, 7.80767907625119652122329569423, 8.894122060608615631780185575065, 9.718475347089371595459280099451, 11.11619810956659061794475203307, 12.01171800047206665236626916843

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