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Properties

Label 2-273-91.80-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.769 + 0.638i$
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  + (0.680 + 2.53i)2-s + i·3-s + (−4.25 + 2.45i)4-s + (0.134 − 0.502i)5-s + (−2.53 + 0.680i)6-s + (−2.56 − 0.650i)7-s + (−5.41 − 5.41i)8-s − 9-s + 1.36·10-s + (1.41 + 1.41i)11-s + (−2.45 − 4.25i)12-s + (−2.40 + 2.68i)13-s + (−0.0921 − 6.95i)14-s + (0.502 + 0.134i)15-s + (5.15 − 8.92i)16-s + (2.54 + 4.41i)17-s + ⋯
L(s)  = 1  + (0.481 + 1.79i)2-s + 0.577i·3-s + (−2.12 + 1.22i)4-s + (0.0602 − 0.224i)5-s + (−1.03 + 0.277i)6-s + (−0.969 − 0.245i)7-s + (−1.91 − 1.91i)8-s − 0.333·9-s + 0.432·10-s + (0.426 + 0.426i)11-s + (−0.709 − 1.22i)12-s + (−0.667 + 0.744i)13-s + (−0.0246 − 1.85i)14-s + (0.129 + 0.0347i)15-s + (1.28 − 2.23i)16-s + (0.617 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.385359 - 1.06824i\)
\(L(\frac12)\) \(\approx\) \(0.385359 - 1.06824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.56 + 0.650i)T \)
13 \( 1 + (2.40 - 2.68i)T \)
good2 \( 1 + (-0.680 - 2.53i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.134 + 0.502i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.41 - 1.41i)T + 11iT^{2} \)
17 \( 1 + (-2.54 - 4.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.09 - 5.09i)T + 19iT^{2} \)
23 \( 1 + (4.14 + 2.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.35 + 2.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.23 + 0.867i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.17 + 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.938 - 3.50i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.62 + 2.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0803 - 0.0215i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.85 + 11.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.34 + 1.43i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 0.753iT - 61T^{2} \)
67 \( 1 + (7.68 - 7.68i)T - 67iT^{2} \)
71 \( 1 + (1.33 + 5.00i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.551 - 2.05i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.17 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.09 + 7.09i)T + 83iT^{2} \)
89 \( 1 + (-3.53 - 13.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.28 + 0.880i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.72688139842461205449833654016, −11.96039275848229938017317252925, −10.01099076666086399640447176044, −9.526927411919552032875033202601, −8.402900716265718084055218036671, −7.43184626209185087319824180564, −6.43306797881578940357529356237, −5.63244501925691768831613064284, −4.43362470747564703807052925288, −3.58593011013778786236214192310, 0.78268490726918708268910503606, 2.69374019377366335864748871356, 3.24154710466591523397275332902, 4.89404972031988243355483558793, 5.96922052977757582374431803638, 7.39671779711409755324342967763, 8.987552478761303283095544540414, 9.679965327077916443828226657698, 10.51537937072293469339185056314, 11.68903201304394591843663878743

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