Properties

Label 2-273-91.19-c1-0-1
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: $\chi_{273} (19, \cdot )$
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

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

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