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Properties

Label 2-273-39.8-c1-0-21
Degree $2$
Conductor $273$
Sign $0.670 + 0.741i$
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  + (1.48 − 1.48i)2-s + (1.53 + 0.809i)3-s − 2.42i·4-s + (−0.854 + 0.854i)5-s + (3.48 − 1.07i)6-s + (0.707 − 0.707i)7-s + (−0.634 − 0.634i)8-s + (1.69 + 2.47i)9-s + 2.54i·10-s + (−0.183 − 0.183i)11-s + (1.96 − 3.71i)12-s + (−2.40 − 2.68i)13-s − 2.10i·14-s + (−1.99 + 0.616i)15-s + 2.96·16-s − 7.60·17-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)2-s + (0.884 + 0.467i)3-s − 1.21i·4-s + (−0.382 + 0.382i)5-s + (1.42 − 0.438i)6-s + (0.267 − 0.267i)7-s + (−0.224 − 0.224i)8-s + (0.563 + 0.826i)9-s + 0.803i·10-s + (−0.0554 − 0.0554i)11-s + (0.566 − 1.07i)12-s + (−0.666 − 0.745i)13-s − 0.562i·14-s + (−0.516 + 0.159i)15-s + 0.741·16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.670 + 0.741i$
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: \((2,\ 273,\ (\ :1/2),\ 0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35532 - 1.04569i\)
\(L(\frac12)\) \(\approx\) \(2.35532 - 1.04569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 - 0.809i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (2.40 + 2.68i)T \)
good2 \( 1 + (-1.48 + 1.48i)T - 2iT^{2} \)
5 \( 1 + (0.854 - 0.854i)T - 5iT^{2} \)
11 \( 1 + (0.183 + 0.183i)T + 11iT^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + (4.33 + 4.33i)T + 19iT^{2} \)
23 \( 1 - 0.600T + 23T^{2} \)
29 \( 1 + 2.36iT - 29T^{2} \)
31 \( 1 + (-6.76 - 6.76i)T + 31iT^{2} \)
37 \( 1 + (2.20 - 2.20i)T - 37iT^{2} \)
41 \( 1 + (-2.79 + 2.79i)T - 41iT^{2} \)
43 \( 1 - 2.48iT - 43T^{2} \)
47 \( 1 + (5.78 + 5.78i)T + 47iT^{2} \)
53 \( 1 - 5.27iT - 53T^{2} \)
59 \( 1 + (-7.52 - 7.52i)T + 59iT^{2} \)
61 \( 1 + 0.457T + 61T^{2} \)
67 \( 1 + (5.51 + 5.51i)T + 67iT^{2} \)
71 \( 1 + (-9.60 + 9.60i)T - 71iT^{2} \)
73 \( 1 + (-4.31 + 4.31i)T - 73iT^{2} \)
79 \( 1 + 4.20T + 79T^{2} \)
83 \( 1 + (0.439 - 0.439i)T - 83iT^{2} \)
89 \( 1 + (3.82 + 3.82i)T + 89iT^{2} \)
97 \( 1 + (-5.33 - 5.33i)T + 97iT^{2} \)
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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

−11.73667125447120349598686541774, −10.82287854595217627582015453241, −10.37779418125062666643768577287, −9.038817874894698948966557681195, −8.014137819731075019490949306725, −6.82336178734995439195082153961, −4.96357741639806334382349829679, −4.31907326061106914449290055465, −3.14020630769958923458377523977, −2.23462794010422435791085670837, 2.25787815656650149222539342521, 4.03580859664910497041547154181, 4.65886748098819869500023189393, 6.23780092569875326577837697416, 6.94043493827132621803297873178, 8.034293161148653910578435610449, 8.663997790578583332466449008762, 9.906609791997304580012333306660, 11.50322491808310823701243298388, 12.55122860371349215042545175498

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