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Properties

Label 2-273-13.12-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.0805 - 0.996i$
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  + 1.29i·2-s − 3-s + 0.334·4-s − 1.33i·5-s − 1.29i·6-s + i·7-s + 3.01i·8-s + 9-s + 1.72·10-s + 4.88i·11-s − 0.334·12-s + (3.59 − 0.290i)13-s − 1.29·14-s + 1.33i·15-s − 3.21·16-s + 4.58·17-s + ⋯
L(s)  = 1  + 0.912i·2-s − 0.577·3-s + 0.167·4-s − 0.596i·5-s − 0.526i·6-s + 0.377i·7-s + 1.06i·8-s + 0.333·9-s + 0.544·10-s + 1.47i·11-s − 0.0965·12-s + (0.996 − 0.0805i)13-s − 0.344·14-s + 0.344i·15-s − 0.804·16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0805 - 0.996i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0805 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.829207 + 0.898943i\)
\(L(\frac12)\) \(\approx\) \(0.829207 + 0.898943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - iT \)
13 \( 1 + (-3.59 + 0.290i)T \)
good2 \( 1 - 1.29iT - 2T^{2} \)
5 \( 1 + 1.33iT - 5T^{2} \)
11 \( 1 - 4.88iT - 11T^{2} \)
17 \( 1 - 4.58T + 17T^{2} \)
19 \( 1 - 2.96iT - 19T^{2} \)
23 \( 1 + 6.13T + 23T^{2} \)
29 \( 1 + 7.63T + 29T^{2} \)
31 \( 1 + 5.63iT - 31T^{2} \)
37 \( 1 + 9.76iT - 37T^{2} \)
41 \( 1 - 3.69iT - 41T^{2} \)
43 \( 1 - 9.63T + 43T^{2} \)
47 \( 1 + 5.27iT - 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 5.50iT - 67T^{2} \)
71 \( 1 + 4.88iT - 71T^{2} \)
73 \( 1 - 4.45iT - 73T^{2} \)
79 \( 1 + 3.54T + 79T^{2} \)
83 \( 1 - 5.27iT - 83T^{2} \)
89 \( 1 + 9.94iT - 89T^{2} \)
97 \( 1 + 18.2iT - 97T^{2} \)
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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

−12.22609197447092347763668712708, −11.33415209979723446586040572135, −10.22725342752357392760793905014, −9.176036876797502361463801934899, −7.981958361200105561781584318346, −7.27916820149496197138447423523, −5.98948727477099169452092171609, −5.46574611784435309914614765565, −4.11542342142085977188427012875, −1.89991757754696464948889841600, 1.13382989259055541112227494903, 3.00273709224930792780520742132, 3.90085870543605897686305727419, 5.72958120324261592209655112441, 6.54465153365751015014753326542, 7.65957611228186134236810493365, 9.017971186718983760595319926924, 10.31226187478168377812048716304, 10.79891655490311148453729235955, 11.46166985465251710283019630050

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