Properties

Label 2-273-91.81-c1-0-1
Degree $2$
Conductor $273$
Sign $0.0121 - 0.999i$
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  + (−1.27 + 2.20i)2-s − 3-s + (−2.25 − 3.90i)4-s + (−1.39 − 2.41i)5-s + (1.27 − 2.20i)6-s + (2.06 + 1.64i)7-s + 6.39·8-s + 9-s + 7.10·10-s − 2.76·11-s + (2.25 + 3.90i)12-s + (2.99 + 2.01i)13-s + (−6.28 + 2.46i)14-s + (1.39 + 2.41i)15-s + (−3.64 + 6.31i)16-s + (2.94 + 5.09i)17-s + ⋯
L(s)  = 1  + (−0.901 + 1.56i)2-s − 0.577·3-s + (−1.12 − 1.95i)4-s + (−0.623 − 1.07i)5-s + (0.520 − 0.901i)6-s + (0.782 + 0.623i)7-s + 2.25·8-s + 0.333·9-s + 2.24·10-s − 0.834·11-s + (0.650 + 1.12i)12-s + (0.830 + 0.557i)13-s + (−1.67 + 0.659i)14-s + (0.359 + 0.623i)15-s + (−0.910 + 1.57i)16-s + (0.713 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0121 - 0.999i$
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: \((2,\ 273,\ (\ :1/2),\ 0.0121 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431848 + 0.426653i\)
\(L(\frac12)\) \(\approx\) \(0.431848 + 0.426653i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 - 1.64i)T \)
13 \( 1 + (-2.99 - 2.01i)T \)
good2 \( 1 + (1.27 - 2.20i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.39 + 2.41i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.76T + 11T^{2} \)
17 \( 1 + (-2.94 - 5.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 + (-3.67 + 6.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.56 - 2.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.93 + 3.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.92 - 5.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.24 - 5.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.99 + 5.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.95 - 6.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.34 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 + 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 + (1.57 - 2.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.80 - 8.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.88 + 3.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.82T + 83T^{2} \)
89 \( 1 + (0.877 - 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.48 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.12873437580344697429471588176, −11.01987338023396426462232494892, −9.978632817811907214460260386625, −8.614782916452071288853013337872, −8.453607945404732101975833692809, −7.44160104858685142671012073523, −6.16600630151181318877119717577, −5.31289645757208769198793449457, −4.50904119344430882767035030232, −1.08565501318001487405422450949, 0.957682406861015008808836567720, 2.84320353025381840407529642504, 3.78420615863969525406118893547, 5.29476847066408170671095196511, 7.39203093402740641182157004070, 7.69692726948574936475048148797, 9.094404711962505681069237320448, 10.35758078378880875082813027271, 10.66751418287245070069376041164, 11.49330255239039089626241610686

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