Properties

Label 2-273-273.272-c1-0-31
Degree $2$
Conductor $273$
Sign $-0.757 + 0.652i$
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  + 1.10·2-s + (−0.669 − 1.59i)3-s − 0.773·4-s − 1.62i·5-s + (−0.741 − 1.76i)6-s + (−2.53 − 0.762i)7-s − 3.07·8-s + (−2.10 + 2.13i)9-s − 1.80i·10-s + 2.37·11-s + (0.517 + 1.23i)12-s + (−2.05 − 2.96i)13-s + (−2.80 − 0.844i)14-s + (−2.59 + 1.08i)15-s − 1.85·16-s + 5.91·17-s + ⋯
L(s)  = 1  + 0.783·2-s + (−0.386 − 0.922i)3-s − 0.386·4-s − 0.727i·5-s + (−0.302 − 0.722i)6-s + (−0.957 − 0.288i)7-s − 1.08·8-s + (−0.701 + 0.712i)9-s − 0.569i·10-s + 0.715·11-s + (0.149 + 0.356i)12-s + (−0.570 − 0.821i)13-s + (−0.749 − 0.225i)14-s + (−0.670 + 0.281i)15-s − 0.463·16-s + 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.757 + 0.652i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.757 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350826 - 0.944972i\)
\(L(\frac12)\) \(\approx\) \(0.350826 - 0.944972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 + 1.59i)T \)
7 \( 1 + (2.53 + 0.762i)T \)
13 \( 1 + (2.05 + 2.96i)T \)
good2 \( 1 - 1.10T + 2T^{2} \)
5 \( 1 + 1.62iT - 5T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 + 2.16T + 19T^{2} \)
23 \( 1 + 7.30iT - 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 + 3.64T + 31T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 - 9.20iT - 47T^{2} \)
53 \( 1 - 12.3iT - 53T^{2} \)
59 \( 1 + 9.86iT - 59T^{2} \)
61 \( 1 + 2.47iT - 61T^{2} \)
67 \( 1 - 5.91iT - 67T^{2} \)
71 \( 1 + 8.43T + 71T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 5.44iT - 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 - 4.17T + 97T^{2} \)
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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

−12.24759310559613651091572580795, −10.74267342775044204825674380620, −9.556334120050000241027044726469, −8.617669682721971534411793708308, −7.47019813037778716128529602977, −6.25324061116448869766955962683, −5.51860923160517329754444380872, −4.32779521763770068838646525701, −2.93482945274692649652574652379, −0.64454107573998933261478041046, 3.14997844355417448422673633495, 3.81269790639769087346894523835, 5.10908230799453036632586466362, 6.03000289354168689150072914873, 6.95746756867881999667131478047, 8.798567561668162990574362906471, 9.601214589620073150380732091885, 10.21505652168844397012303627505, 11.64527636152007731340453883512, 12.07657569800867028156535656914

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