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Properties

Label 2-273-273.17-c1-0-8
Degree $2$
Conductor $273$
Sign $0.280 + 0.959i$
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  − 2.13·2-s + (−1.59 − 0.669i)3-s + 2.56·4-s + (−3.07 + 1.77i)5-s + (3.41 + 1.42i)6-s + (−1.82 + 1.91i)7-s − 1.19·8-s + (2.10 + 2.13i)9-s + (6.56 − 3.78i)10-s + (0.843 + 1.46i)11-s + (−4.09 − 1.71i)12-s + (−0.905 − 3.49i)13-s + (3.88 − 4.10i)14-s + (6.09 − 0.776i)15-s − 2.56·16-s − 2.50·17-s + ⋯
L(s)  = 1  − 1.51·2-s + (−0.922 − 0.386i)3-s + 1.28·4-s + (−1.37 + 0.793i)5-s + (1.39 + 0.583i)6-s + (−0.688 + 0.725i)7-s − 0.423·8-s + (0.701 + 0.713i)9-s + (2.07 − 1.19i)10-s + (0.254 + 0.440i)11-s + (−1.18 − 0.494i)12-s + (−0.251 − 0.967i)13-s + (1.03 − 1.09i)14-s + (1.57 − 0.200i)15-s − 0.641·16-s − 0.608·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128506 - 0.0962807i\)
\(L(\frac12)\) \(\approx\) \(0.128506 - 0.0962807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 + 0.669i)T \)
7 \( 1 + (1.82 - 1.91i)T \)
13 \( 1 + (0.905 + 3.49i)T \)
good2 \( 1 + 2.13T + 2T^{2} \)
5 \( 1 + (3.07 - 1.77i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.843 - 1.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.50T + 17T^{2} \)
19 \( 1 + (1.66 - 2.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.63iT - 23T^{2} \)
29 \( 1 + (-3.85 - 2.22i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.26 + 9.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.55iT - 37T^{2} \)
41 \( 1 + (2.89 + 1.67i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.303 + 0.525i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.95 + 1.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.84 - 2.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.72iT - 59T^{2} \)
61 \( 1 + (-7.83 - 4.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.411 + 0.237i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.84 + 6.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.61 - 2.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.65 + 6.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 + 5.07iT - 89T^{2} \)
97 \( 1 + (-4.00 - 6.93i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63365624155469752758644075889, −10.49709904691717435083834419155, −10.15227043467592073443894404444, −8.645894256751302411307708116834, −7.86985849333122210707843620131, −6.97805980963793940237033990606, −6.23827511622316193527620484821, −4.41554081344354661686369503597, −2.56066339480981715126434425244, −0.30694900058343763412905500404, 0.944121610664408250031002936123, 3.83858971613208656090632042970, 4.77050012019619368940579759758, 6.64192096844633530910409073196, 7.25967104448606775322095648743, 8.462347324634017137729077293558, 9.243447345387820270552536342938, 10.10191225607651745282453153883, 11.17501726738415206440797562396, 11.58475739193657514259308040049

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