Properties

Label 2-273-91.24-c1-0-11
Degree $2$
Conductor $273$
Sign $-0.339 + 0.940i$
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  + (−0.511 + 0.137i)2-s + i·3-s + (−1.48 + 0.859i)4-s + (−2.03 − 0.545i)5-s + (−0.137 − 0.511i)6-s + (0.917 − 2.48i)7-s + (1.39 − 1.39i)8-s − 9-s + 1.11·10-s + (−2.03 + 2.03i)11-s + (−0.859 − 1.48i)12-s + (−2.80 − 2.27i)13-s + (−0.129 + 1.39i)14-s + (0.545 − 2.03i)15-s + (1.19 − 2.07i)16-s + (−1.64 − 2.85i)17-s + ⋯
L(s)  = 1  + (−0.361 + 0.0969i)2-s + 0.577i·3-s + (−0.744 + 0.429i)4-s + (−0.911 − 0.244i)5-s + (−0.0559 − 0.208i)6-s + (0.346 − 0.937i)7-s + (0.492 − 0.492i)8-s − 0.333·9-s + 0.353·10-s + (−0.614 + 0.614i)11-s + (−0.248 − 0.429i)12-s + (−0.776 − 0.629i)13-s + (−0.0345 + 0.373i)14-s + (0.140 − 0.526i)15-s + (0.299 − 0.518i)16-s + (−0.399 − 0.692i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.339 + 0.940i$
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: \((2,\ 273,\ (\ :1/2),\ -0.339 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165366 - 0.235487i\)
\(L(\frac12)\) \(\approx\) \(0.165366 - 0.235487i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.917 + 2.48i)T \)
13 \( 1 + (2.80 + 2.27i)T \)
good2 \( 1 + (0.511 - 0.137i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.03 + 0.545i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.03 - 2.03i)T - 11iT^{2} \)
17 \( 1 + (1.64 + 2.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.78 + 4.78i)T - 19iT^{2} \)
23 \( 1 + (6.79 + 3.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.677 - 1.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.71 - 6.38i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.05 - 3.93i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.61 + 1.77i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.36 + 2.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.947 - 3.53i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.87 - 6.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.737 + 2.75i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.63iT - 61T^{2} \)
67 \( 1 + (1.33 + 1.33i)T + 67iT^{2} \)
71 \( 1 + (9.87 - 2.64i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.80 + 1.82i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.13 + 1.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.97 + 3.97i)T - 83iT^{2} \)
89 \( 1 + (-11.9 + 3.19i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.94 + 14.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.64595031727539354211231340377, −10.38037861564429822164946726096, −9.832092707267006338829878608752, −8.631121608153468018441574707734, −7.79617047822287870174712813633, −7.14863030361726471215347119714, −4.89630529691891196445488547162, −4.52904403846858530441430458521, −3.20225899484696934416755545910, −0.24842674189567236805730497158, 1.95715984636027796023124746700, 3.75430262859385268222145888721, 5.17910003470037117665575960667, 6.11418104369095144464867581662, 7.85823949947690488347676104792, 8.058570656912045795900613207021, 9.288608813424864184974422870012, 10.21017608140511300847224082852, 11.53865426625393739734083636721, 11.87469678664402078712874363504

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