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Properties

Label 2-273-13.12-c1-0-9
Degree $2$
Conductor $273$
Sign $0.999 + 0.0139i$
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  + 2.60i·2-s − 3-s − 4.78·4-s − 3.78i·5-s − 2.60i·6-s − i·7-s − 7.26i·8-s + 9-s + 9.86·10-s + 2.55i·11-s + 4.78·12-s + (0.0504 − 3.60i)13-s + 2.60·14-s + 3.78i·15-s + 9.34·16-s − 3.21·17-s + ⋯
L(s)  = 1  + 1.84i·2-s − 0.577·3-s − 2.39·4-s − 1.69i·5-s − 1.06i·6-s − 0.377i·7-s − 2.56i·8-s + 0.333·9-s + 3.11·10-s + 0.770i·11-s + 1.38·12-s + (0.0139 − 0.999i)13-s + 0.696·14-s + 0.977i·15-s + 2.33·16-s − 0.778·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0139i$
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: \((2,\ 273,\ (\ :1/2),\ 0.999 + 0.0139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689165 - 0.00482116i\)
\(L(\frac12)\) \(\approx\) \(0.689165 - 0.00482116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0504 + 3.60i)T \)
good2 \( 1 - 2.60iT - 2T^{2} \)
5 \( 1 + 3.78iT - 5T^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
17 \( 1 + 3.21T + 17T^{2} \)
19 \( 1 + 8.44iT - 19T^{2} \)
23 \( 1 - 3.97T + 23T^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 - 0.868iT - 31T^{2} \)
37 \( 1 + 5.10iT - 37T^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 - 1.33T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 + 4.10T + 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 2.55iT - 71T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 + 1.23T + 79T^{2} \)
83 \( 1 + 11.0iT - 83T^{2} \)
89 \( 1 - 5.52iT - 89T^{2} \)
97 \( 1 - 5.65iT - 97T^{2} \)
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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

−12.40501148830028607896682818469, −10.84717425481544198818309178746, −9.333585296183280485376945106035, −8.971745893211077903057079126058, −7.78194743853113641228270778780, −7.02823843531887891378706487479, −5.80566514495652243480370975897, −4.89788523456000695994481810766, −4.43744846945057481022249463891, −0.58587456943888545848658042567, 1.92936211941682309606075020867, 3.14672796628683823510044043828, 4.09932386311301587863076699306, 5.70500596989721803795057690701, 6.83402604417269888314679100406, 8.394630935835769150844302279473, 9.617578497998867383954365145670, 10.34342225116925440242208628366, 11.20624085712223372363836853535, 11.46952878419035185351105067904

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