Properties

Label 2-273-273.68-c1-0-22
Degree $2$
Conductor $273$
Sign $0.879 + 0.475i$
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  − 0.771i·2-s + (0.714 + 1.57i)3-s + 1.40·4-s + (0.907 − 1.57i)5-s + (1.21 − 0.551i)6-s + (0.574 − 2.58i)7-s − 2.62i·8-s + (−1.97 + 2.25i)9-s + (−1.21 − 0.699i)10-s + (0.504 + 0.291i)11-s + (1.00 + 2.21i)12-s + (−3.53 + 0.728i)13-s + (−1.99 − 0.442i)14-s + (3.12 + 0.308i)15-s + 0.785·16-s + 0.958·17-s + ⋯
L(s)  = 1  − 0.545i·2-s + (0.412 + 0.910i)3-s + 0.702·4-s + (0.405 − 0.703i)5-s + (0.496 − 0.225i)6-s + (0.217 − 0.976i)7-s − 0.928i·8-s + (−0.659 + 0.751i)9-s + (−0.383 − 0.221i)10-s + (0.152 + 0.0878i)11-s + (0.289 + 0.640i)12-s + (−0.979 + 0.201i)13-s + (−0.532 − 0.118i)14-s + (0.807 + 0.0796i)15-s + 0.196·16-s + 0.232·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70231 - 0.430887i\)
\(L(\frac12)\) \(\approx\) \(1.70231 - 0.430887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.714 - 1.57i)T \)
7 \( 1 + (-0.574 + 2.58i)T \)
13 \( 1 + (3.53 - 0.728i)T \)
good2 \( 1 + 0.771iT - 2T^{2} \)
5 \( 1 + (-0.907 + 1.57i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.504 - 0.291i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.958T + 17T^{2} \)
19 \( 1 + (1.66 - 0.962i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.73iT - 23T^{2} \)
29 \( 1 + (-4.26 + 2.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.83 - 1.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.79T + 37T^{2} \)
41 \( 1 + (1.73 + 2.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.367 - 0.636i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.49 - 6.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.08 - 2.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (3.31 - 1.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.07 + 8.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.73 - 5.61i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.30 - 4.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.35 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.45T + 83T^{2} \)
89 \( 1 + 7.06T + 89T^{2} \)
97 \( 1 + (7.89 + 4.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.63308860890627626494430499195, −10.81039186560194679097594881554, −9.921017058945047112815173486886, −9.398584680338849737543932231852, −8.011565475873222251187374995800, −7.04450481948069315228969770716, −5.49099780892648121211514559603, −4.37307562793610968155260409007, −3.24791726786017806701783552557, −1.69751445256779398527435515151, 2.17675708174376192591841226528, 2.83961972223350219822608454837, 5.23094486932544278648011985275, 6.39794659349696246265923172503, 6.83104248847118274281436376043, 8.016454851803795218549924607436, 8.726434025287530582279851231312, 10.09932850279364127823669365604, 11.18869143158564788263537451037, 12.12839786059630484344604481880

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