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Properties

Label 2-273-91.9-c1-0-15
Degree $2$
Conductor $273$
Sign $0.808 - 0.588i$
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  + (0.828 + 1.43i)2-s + 3-s + (−0.372 + 0.644i)4-s + (1.05 − 1.81i)5-s + (0.828 + 1.43i)6-s + (−1.14 − 2.38i)7-s + 2.07·8-s + 9-s + 3.47·10-s − 0.304·11-s + (−0.372 + 0.644i)12-s + (−0.494 + 3.57i)13-s + (2.47 − 3.61i)14-s + (1.05 − 1.81i)15-s + (2.46 + 4.27i)16-s + (−2.90 + 5.03i)17-s + ⋯
L(s)  = 1  + (0.585 + 1.01i)2-s + 0.577·3-s + (−0.186 + 0.322i)4-s + (0.469 − 0.813i)5-s + (0.338 + 0.585i)6-s + (−0.433 − 0.901i)7-s + 0.735·8-s + 0.333·9-s + 1.10·10-s − 0.0917·11-s + (−0.107 + 0.186i)12-s + (−0.137 + 0.990i)13-s + (0.660 − 0.967i)14-s + (0.271 − 0.469i)15-s + (0.616 + 1.06i)16-s + (−0.704 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03442 + 0.661985i\)
\(L(\frac12)\) \(\approx\) \(2.03442 + 0.661985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 + 2.38i)T \)
13 \( 1 + (0.494 - 3.57i)T \)
good2 \( 1 + (-0.828 - 1.43i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.05 + 1.81i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.304T + 11T^{2} \)
17 \( 1 + (2.90 - 5.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.48T + 19T^{2} \)
23 \( 1 + (3.00 + 5.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.09 + 1.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.51 - 6.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0524 - 0.0908i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.00 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.03 - 6.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.30 + 7.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.10 + 1.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.84 + 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + (0.149 + 0.259i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.96 + 5.14i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.54 + 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.33T + 83T^{2} \)
89 \( 1 + (1.54 + 2.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.19 + 5.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.60715806164513010453020768137, −10.82060659261966813933649974670, −10.07986018141309750699737617403, −8.829548169692513646707052959214, −8.096867697255131535769268588132, −6.74268140627377089655778672598, −6.27842605603461550848017079225, −4.67769655888298787655017273104, −4.09124017987606895237207739919, −1.85282084849716253894680256093, 2.35973197015964518609931394506, 2.78825200481499339720208971648, 4.14051937329643110585982072131, 5.58597872500376765741495689057, 6.81401769830529844690181676607, 7.987899409368672179574462922966, 9.209003036860649002866561567411, 10.15957164850166040836737277700, 10.87867608970012166039784648710, 11.91370249594126784781464180811

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