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Properties

Label 2-273-91.33-c1-0-9
Degree $2$
Conductor $273$
Sign $0.986 + 0.164i$
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  + (0.255 − 0.954i)2-s + i·3-s + (0.886 + 0.511i)4-s + (−0.244 − 0.912i)5-s + (0.954 + 0.255i)6-s + (2.46 + 0.953i)7-s + (2.11 − 2.11i)8-s − 9-s − 0.933·10-s + (−4.64 + 4.64i)11-s + (−0.511 + 0.886i)12-s + (3.43 − 1.08i)13-s + (1.54 − 2.11i)14-s + (0.912 − 0.244i)15-s + (−0.452 − 0.783i)16-s + (1.86 − 3.23i)17-s + ⋯
L(s)  = 1  + (0.180 − 0.674i)2-s + 0.577i·3-s + (0.443 + 0.255i)4-s + (−0.109 − 0.407i)5-s + (0.389 + 0.104i)6-s + (0.932 + 0.360i)7-s + (0.746 − 0.746i)8-s − 0.333·9-s − 0.295·10-s + (−1.40 + 1.40i)11-s + (−0.147 + 0.255i)12-s + (0.953 − 0.299i)13-s + (0.411 − 0.564i)14-s + (0.235 − 0.0631i)15-s + (−0.113 − 0.195i)16-s + (0.452 − 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66308 - 0.137649i\)
\(L(\frac12)\) \(\approx\) \(1.66308 - 0.137649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-2.46 - 0.953i)T \)
13 \( 1 + (-3.43 + 1.08i)T \)
good2 \( 1 + (-0.255 + 0.954i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.244 + 0.912i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.64 - 4.64i)T - 11iT^{2} \)
17 \( 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.889 + 0.889i)T - 19iT^{2} \)
23 \( 1 + (-0.202 + 0.116i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.32 - 4.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.35 + 1.96i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.85 + 1.30i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.49 + 9.29i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (10.4 - 6.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.10 - 0.563i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.04 - 3.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.20 - 1.93i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 2.45iT - 61T^{2} \)
67 \( 1 + (7.19 + 7.19i)T + 67iT^{2} \)
71 \( 1 + (-0.433 + 1.61i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.10 - 7.85i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.942 - 1.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.95 + 9.95i)T - 83iT^{2} \)
89 \( 1 + (-1.05 + 3.94i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.41 + 0.647i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.86582766660002832437267147795, −10.87307764281289242390839672659, −10.36160968196845484440579740340, −9.137275923370377633234629199161, −8.005172351890221910176295571191, −7.19004270150525056238007098190, −5.36311316048829704247414055757, −4.62519892728790844576296049460, −3.21140394505677214239412224676, −1.88841515550666415936070008164, 1.63256500312105461974510943636, 3.31622161260618299224365127442, 5.19157826265915902476904035133, 5.94504968063357219654718304977, 7.01093075788439122801349011946, 7.973177706642439287862565397642, 8.460982774520465917894112685302, 10.48867813147842883188185797983, 10.95922466744326873981465912416, 11.68947082156003867957537095338

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