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Properties

Label 2-273-39.5-c1-0-7
Degree $2$
Conductor $273$
Sign $0.447 - 0.894i$
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 + i)2-s + (−1.70 − 0.292i)3-s + (2.70 + 2.70i)5-s + (−1.41 − 2i)6-s + (−0.707 − 0.707i)7-s + (2 − 2i)8-s + (2.82 + i)9-s + 5.41i·10-s + (−1.41 + 1.41i)11-s + (0.707 + 3.53i)13-s − 1.41i·14-s + (−3.82 − 5.41i)15-s + 4·16-s + 4·17-s + (1.82 + 3.82i)18-s + (−1.87 + 1.87i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.985 − 0.169i)3-s + (1.21 + 1.21i)5-s + (−0.577 − 0.816i)6-s + (−0.267 − 0.267i)7-s + (0.707 − 0.707i)8-s + (0.942 + 0.333i)9-s + 1.71i·10-s + (−0.426 + 0.426i)11-s + (0.196 + 0.980i)13-s − 0.377i·14-s + (−0.988 − 1.39i)15-s + 16-s + 0.970·17-s + (0.430 + 0.902i)18-s + (−0.430 + 0.430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.447 - 0.894i$
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: \((2,\ 273,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38894 + 0.858154i\)
\(L(\frac12)\) \(\approx\) \(1.38894 + 0.858154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.292i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.707 - 3.53i)T \)
good2 \( 1 + (-1 - i)T + 2iT^{2} \)
5 \( 1 + (-2.70 - 2.70i)T + 5iT^{2} \)
11 \( 1 + (1.41 - 1.41i)T - 11iT^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (1.87 - 1.87i)T - 19iT^{2} \)
23 \( 1 + 1.82T + 23T^{2} \)
29 \( 1 + 6.65iT - 29T^{2} \)
31 \( 1 + (-2.94 + 2.94i)T - 31iT^{2} \)
37 \( 1 + (6.41 + 6.41i)T + 37iT^{2} \)
41 \( 1 + (-0.242 - 0.242i)T + 41iT^{2} \)
43 \( 1 + 6.65iT - 43T^{2} \)
47 \( 1 + (1.87 - 1.87i)T - 47iT^{2} \)
53 \( 1 + 12.3iT - 53T^{2} \)
59 \( 1 + (-1.41 + 1.41i)T - 59iT^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + (9.07 - 9.07i)T - 67iT^{2} \)
71 \( 1 + (-11.2 - 11.2i)T + 71iT^{2} \)
73 \( 1 + (-0.121 - 0.121i)T + 73iT^{2} \)
79 \( 1 - 5.82T + 79T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + 83iT^{2} \)
89 \( 1 + (-5.87 + 5.87i)T - 89iT^{2} \)
97 \( 1 + (7.29 - 7.29i)T - 97iT^{2} \)
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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

−12.21032478155941524049775560899, −10.99124813111117394222760838305, −10.15168650001607235257657089609, −9.773405525951887454144814736578, −7.53768224808193586969351667511, −6.71780752422484728308714635348, −6.11012138523813880005662364968, −5.35027148047421015330174439788, −3.99396257667187070314564170560, −1.93813770050050353877005677327, 1.40167872400662859160517586924, 3.13835238599765536614706366613, 4.77143753746381227798769931741, 5.33761085377088589800288187358, 6.21010915021482259424670481310, 7.955129482345064531248970639130, 9.076305710598061974172431082014, 10.22032595337653246737341858755, 10.78694030857447550688423134976, 12.19976827111263639908427954809

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