Properties

Label 2-273-91.81-c1-0-10
Degree $2$
Conductor $273$
Sign $0.923 - 0.383i$
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  + (0.130 − 0.226i)2-s + 3-s + (0.965 + 1.67i)4-s + (0.708 + 1.22i)5-s + (0.130 − 0.226i)6-s + (−0.675 − 2.55i)7-s + 1.02·8-s + 9-s + 0.370·10-s − 0.845·11-s + (0.965 + 1.67i)12-s + (1.58 + 3.24i)13-s + (−0.666 − 0.181i)14-s + (0.708 + 1.22i)15-s + (−1.79 + 3.11i)16-s + (−2.54 − 4.41i)17-s + ⋯
L(s)  = 1  + (0.0923 − 0.159i)2-s + 0.577·3-s + (0.482 + 0.836i)4-s + (0.316 + 0.548i)5-s + (0.0533 − 0.0923i)6-s + (−0.255 − 0.966i)7-s + 0.363·8-s + 0.333·9-s + 0.117·10-s − 0.254·11-s + (0.278 + 0.482i)12-s + (0.438 + 0.898i)13-s + (−0.178 − 0.0484i)14-s + (0.182 + 0.316i)15-s + (−0.449 + 0.778i)16-s + (−0.618 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.923 - 0.383i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75167 + 0.348793i\)
\(L(\frac12)\) \(\approx\) \(1.75167 + 0.348793i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.675 + 2.55i)T \)
13 \( 1 + (-1.58 - 3.24i)T \)
good2 \( 1 + (-0.130 + 0.226i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.708 - 1.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.845T + 11T^{2} \)
17 \( 1 + (2.54 + 4.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (4.60 - 7.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.02 + 6.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.19 + 3.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.69 + 8.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.16 + 8.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.33 - 9.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.80 - 3.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.08 - 1.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.01 + 8.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 3.33T + 61T^{2} \)
67 \( 1 + 3.50T + 67T^{2} \)
71 \( 1 + (6.45 - 11.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0588 - 0.101i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.76 + 9.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.17T + 83T^{2} \)
89 \( 1 + (-3.11 + 5.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.165 - 0.285i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.74819805387375723212642695745, −11.21040546602178243449124656115, −10.02622193883667025948006696222, −9.219054382779333777590297078196, −7.78303466896036055854528531558, −7.28384038822198843289724536508, −6.23664797351550482716364381558, −4.30470833673363978248386154054, −3.37438223260588294279795404254, −2.14225823026994926383122550886, 1.66330590309698600778714559512, 3.02557866324811856953312866221, 4.87389918649985961085425621729, 5.79688746272889469161770185296, 6.70795208188123804435016759027, 8.198449573464112317656880751557, 8.887310082623940161811925505053, 9.987506419143013790947469384807, 10.67969999694051881481767540643, 11.95645130257382212136186922293

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