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Properties

Label 2-273-21.5-c1-0-28
Degree $2$
Conductor $273$
Sign $-0.222 + 0.974i$
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  + (−0.636 − 0.367i)2-s + (1.70 − 0.328i)3-s + (−0.730 − 1.26i)4-s + (1.03 − 1.78i)5-s + (−1.20 − 0.416i)6-s + (−0.617 − 2.57i)7-s + 2.54i·8-s + (2.78 − 1.11i)9-s + (−1.31 + 0.757i)10-s + (−4.03 + 2.32i)11-s + (−1.65 − 1.91i)12-s + i·13-s + (−0.552 + 1.86i)14-s + (1.16 − 3.37i)15-s + (−0.525 + 0.910i)16-s + (−2.37 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.449 − 0.259i)2-s + (0.981 − 0.189i)3-s + (−0.365 − 0.632i)4-s + (0.461 − 0.798i)5-s + (−0.491 − 0.169i)6-s + (−0.233 − 0.972i)7-s + 0.898i·8-s + (0.928 − 0.372i)9-s + (−0.414 + 0.239i)10-s + (−1.21 + 0.701i)11-s + (−0.478 − 0.551i)12-s + 0.277i·13-s + (−0.147 + 0.498i)14-s + (0.301 − 0.871i)15-s + (−0.131 + 0.227i)16-s + (−0.575 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.222 + 0.974i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.222 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783826 - 0.982714i\)
\(L(\frac12)\) \(\approx\) \(0.783826 - 0.982714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.328i)T \)
7 \( 1 + (0.617 + 2.57i)T \)
13 \( 1 - iT \)
good2 \( 1 + (0.636 + 0.367i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.03 + 1.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.03 - 2.32i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.37 + 4.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.18 - 1.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.58 - 3.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.76iT - 29T^{2} \)
31 \( 1 + (-4.49 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.73 - 4.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.99T + 41T^{2} \)
43 \( 1 - 8.41T + 43T^{2} \)
47 \( 1 + (-1.09 + 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.754 - 0.435i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.620 - 1.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.38 - 5.41i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.1iT - 71T^{2} \)
73 \( 1 + (6.04 - 3.48i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.36 - 5.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.428T + 83T^{2} \)
89 \( 1 + (1.97 - 3.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.05iT - 97T^{2} \)
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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

−11.49801875883290972427862571290, −10.22572897058698361270557137947, −9.710118596349446138226076964870, −8.988893866201706347852435861284, −7.916089056972252134099217517076, −6.99182774123252090980435484292, −5.29502518041014172003635221445, −4.38704140017462087720861463665, −2.55902144725455905823794567362, −1.10561860471662721509962123082, 2.62188565134885036369653958026, 3.28679201554998329788160337076, 4.99180692470611540743232067011, 6.48278176872150639637110406306, 7.51131188822167116823274175234, 8.604650434661716608980848010308, 8.923689731792601890234844730695, 10.17100147994192895659660556892, 10.84719859406924618798376971274, 12.57376685001046185259832569420

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