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Properties

Label 2-273-91.34-c1-0-8
Degree $2$
Conductor $273$
Sign $-0.0934 + 0.995i$
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  + (−1.22 − 1.22i)2-s + i·3-s + 0.999i·4-s + (−2 + 2i)5-s + (1.22 − 1.22i)6-s + (1 − 2.44i)7-s + (−1.22 + 1.22i)8-s − 9-s + 4.89·10-s + (4.44 − 4.44i)11-s − 0.999·12-s + (−2 − 3i)13-s + (−4.22 + 1.77i)14-s + (−2 − 2i)15-s + 5·16-s + 2·17-s + ⋯
L(s)  = 1  + (−0.866 − 0.866i)2-s + 0.577i·3-s + 0.499i·4-s + (−0.894 + 0.894i)5-s + (0.499 − 0.499i)6-s + (0.377 − 0.925i)7-s + (−0.433 + 0.433i)8-s − 0.333·9-s + 1.54·10-s + (1.34 − 1.34i)11-s − 0.288·12-s + (−0.554 − 0.832i)13-s + (−1.12 + 0.474i)14-s + (−0.516 − 0.516i)15-s + 1.25·16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0934 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0934 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0934 + 0.995i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0934 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447737 - 0.491734i\)
\(L(\frac12)\) \(\approx\) \(0.447737 - 0.491734i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2.44i)T \)
13 \( 1 + (2 + 3i)T \)
good2 \( 1 + (1.22 + 1.22i)T + 2iT^{2} \)
5 \( 1 + (2 - 2i)T - 5iT^{2} \)
11 \( 1 + (-4.44 + 4.44i)T - 11iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (-5.44 + 5.44i)T - 19iT^{2} \)
23 \( 1 + 0.898iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 + (-0.550 + 0.550i)T - 31iT^{2} \)
37 \( 1 + (1.89 - 1.89i)T - 37iT^{2} \)
41 \( 1 + (-4 + 4i)T - 41iT^{2} \)
43 \( 1 - 2.89iT - 43T^{2} \)
47 \( 1 + (6.44 + 6.44i)T + 47iT^{2} \)
53 \( 1 - 7.79T + 53T^{2} \)
59 \( 1 + (0.449 + 0.449i)T + 59iT^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + (-8.34 - 8.34i)T + 67iT^{2} \)
71 \( 1 + (2.44 + 2.44i)T + 71iT^{2} \)
73 \( 1 + (-1.89 - 1.89i)T + 73iT^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 + (-4.44 + 4.44i)T - 83iT^{2} \)
89 \( 1 + (2 + 2i)T + 89iT^{2} \)
97 \( 1 + (-1.89 + 1.89i)T - 97iT^{2} \)
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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

−11.35519702266389448169670766456, −10.82418411023606601414871317734, −9.942491556723721419973196148026, −9.017752054225046733100565310173, −7.979219896661870137615300253636, −7.01867158655110493116961113647, −5.46801939783081782476974494643, −3.78216503385497611169377832204, −3.05217263645611403260673527893, −0.75275537138415115216343060825, 1.55380507360430923360558612234, 3.87328209562343459947022186901, 5.26529520665939108739305043002, 6.55282958961159371465570313225, 7.55869524182978500225695167800, 8.066805716109816110699993791443, 9.254804720720339534875236135128, 9.517281777163213751625976941436, 11.67259929507784629426960809745, 12.13483086447922075359144655109

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