Properties

Label 2-273-7.4-c1-0-10
Degree $2$
Conductor $273$
Sign $0.999 - 0.0375i$
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  + (0.893 + 1.54i)2-s + (0.5 − 0.866i)3-s + (−0.595 + 1.03i)4-s + (−1.71 − 2.97i)5-s + 1.78·6-s + (2.14 − 1.54i)7-s + 1.44·8-s + (−0.499 − 0.866i)9-s + (3.06 − 5.30i)10-s + (−0.797 + 1.38i)11-s + (0.595 + 1.03i)12-s − 13-s + (4.31 + 1.93i)14-s − 3.42·15-s + (2.48 + 4.29i)16-s + (−0.0197 + 0.0341i)17-s + ⋯
L(s)  = 1  + (0.631 + 1.09i)2-s + (0.288 − 0.499i)3-s + (−0.297 + 0.515i)4-s + (−0.766 − 1.32i)5-s + 0.729·6-s + (0.811 − 0.584i)7-s + 0.510·8-s + (−0.166 − 0.288i)9-s + (0.968 − 1.67i)10-s + (−0.240 + 0.416i)11-s + (0.171 + 0.297i)12-s − 0.277·13-s + (1.15 + 0.518i)14-s − 0.885·15-s + (0.620 + 1.07i)16-s + (−0.00478 + 0.00829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0375i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 - 0.0375i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86227 + 0.0350198i\)
\(L(\frac12)\) \(\approx\) \(1.86227 + 0.0350198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.14 + 1.54i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.893 - 1.54i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.71 + 2.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.797 - 1.38i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0197 - 0.0341i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.82 - 3.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.71 - 2.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 + (-3.07 + 5.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.56 - 9.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 - 2.91T + 43T^{2} \)
47 \( 1 + (-2.05 - 3.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.14 + 8.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.59 - 7.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.980 - 1.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.02 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.79T + 71T^{2} \)
73 \( 1 + (-3.35 + 5.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.31 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.34T + 83T^{2} \)
89 \( 1 + (-6.23 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−12.13640168676615713964096951380, −11.29194161363183713978270283610, −9.825861650775622261903383757580, −8.494408882826312373069429519074, −7.75303229396497035883029276855, −7.27572188546545444922427374086, −5.73691288641393157927730712331, −4.81293024830488028577539203418, −3.98505992817005692494580769089, −1.42877496384951945540668940667, 2.36899698147735408895707217990, 3.20946693541262952884735989019, 4.25537256432619811268722132498, 5.41327857564611772143441664073, 7.14397650940326425111816187633, 7.978136149421454100476785625062, 9.263874918646445622551024466472, 10.59591313047792567366255536581, 11.02751008041818519845810548408, 11.63979992230965617694991293977

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