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Properties

Label 2-273-91.9-c1-0-12
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: $\chi_{273} (100, \cdot )$
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

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

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