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Properties

Label 2-273-91.24-c1-0-5
Degree $2$
Conductor $273$
Sign $-0.364 - 0.931i$
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  + (−0.759 + 0.203i)2-s + i·3-s + (−1.19 + 0.690i)4-s + (2.02 + 0.542i)5-s + (−0.203 − 0.759i)6-s + (1.65 + 2.06i)7-s + (1.88 − 1.88i)8-s − 9-s − 1.64·10-s + (−1.72 + 1.72i)11-s + (−0.690 − 1.19i)12-s + (−1.96 − 3.02i)13-s + (−1.67 − 1.22i)14-s + (−0.542 + 2.02i)15-s + (0.335 − 0.581i)16-s + (3.27 + 5.67i)17-s + ⋯
L(s)  = 1  + (−0.537 + 0.143i)2-s + 0.577i·3-s + (−0.598 + 0.345i)4-s + (0.906 + 0.242i)5-s + (−0.0830 − 0.310i)6-s + (0.626 + 0.779i)7-s + (0.664 − 0.664i)8-s − 0.333·9-s − 0.521·10-s + (−0.521 + 0.521i)11-s + (−0.199 − 0.345i)12-s + (−0.546 − 0.837i)13-s + (−0.448 − 0.328i)14-s + (−0.140 + 0.523i)15-s + (0.0839 − 0.145i)16-s + (0.794 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.364 - 0.931i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517004 + 0.757849i\)
\(L(\frac12)\) \(\approx\) \(0.517004 + 0.757849i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 2.06i)T \)
13 \( 1 + (1.96 + 3.02i)T \)
good2 \( 1 + (0.759 - 0.203i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.02 - 0.542i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.72 - 1.72i)T - 11iT^{2} \)
17 \( 1 + (-3.27 - 5.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.66 - 5.66i)T - 19iT^{2} \)
23 \( 1 + (-3.33 - 1.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.122 + 0.457i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.83 + 6.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-11.0 - 2.94i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.810 - 0.467i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.88 + 7.03i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.08 - 1.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.43 + 5.36i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 9.39iT - 61T^{2} \)
67 \( 1 + (-8.15 - 8.15i)T + 67iT^{2} \)
71 \( 1 + (-9.44 + 2.52i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-9.79 + 2.62i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.07 - 1.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.52 - 1.52i)T - 83iT^{2} \)
89 \( 1 + (-6.08 + 1.62i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.17 + 15.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.50979155473612797870857004768, −10.83035599438047767738377379717, −10.12347534488572542196565867042, −9.461049536023252742796938967039, −8.346860145734244992331902982693, −7.75803203116162327726961125457, −5.99458325139672102412337297755, −5.17728708634806499915802100264, −3.85044805182497711785252376563, −2.14023533072444169897684304867, 0.895581836729536866729099994914, 2.35309468383612157546494743611, 4.59690272562627656397289475698, 5.39078207763221320169832312623, 6.81127366548331488025435433496, 7.81666115787639628527348428826, 8.905142542129201476400253180174, 9.564097774299052903039748547103, 10.65323796556066780950159643441, 11.32753258557882759111817365301

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