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Properties

Label 2-273-273.86-c1-0-20
Degree $2$
Conductor $273$
Sign $0.825 - 0.563i$
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  + (0.258 + 0.965i)2-s + (1.10 + 1.33i)3-s + (0.866 − 0.5i)4-s + (−0.883 − 3.29i)5-s + (−1 + 1.41i)6-s + (1.15 − 2.38i)7-s + (2.12 + 2.12i)8-s + (−0.548 + 2.94i)9-s + (2.95 − 1.70i)10-s + (−0.0444 + 0.165i)11-s + (1.62 + 0.599i)12-s + (2 + 3i)13-s + (2.59 + 0.500i)14-s + (3.41 − 4.82i)15-s + (−0.500 + 0.866i)16-s + (−1.91 − 3.31i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.639 + 0.769i)3-s + (0.433 − 0.250i)4-s + (−0.395 − 1.47i)5-s + (−0.408 + 0.577i)6-s + (0.436 − 0.899i)7-s + (0.749 + 0.749i)8-s + (−0.182 + 0.983i)9-s + (0.935 − 0.539i)10-s + (−0.0133 + 0.0499i)11-s + (0.469 + 0.173i)12-s + (0.554 + 0.832i)13-s + (0.694 + 0.133i)14-s + (0.881 − 1.24i)15-s + (−0.125 + 0.216i)16-s + (−0.464 − 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.825 - 0.563i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.825 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78841 + 0.552210i\)
\(L(\frac12)\) \(\approx\) \(1.78841 + 0.552210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.10 - 1.33i)T \)
7 \( 1 + (-1.15 + 2.38i)T \)
13 \( 1 + (-2 - 3i)T \)
good2 \( 1 + (-0.258 - 0.965i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.883 + 3.29i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0444 - 0.165i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.91 + 3.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.49 - 1.20i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.70 - 8.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + iT - 29T^{2} \)
31 \( 1 + (-1.24 + 4.66i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.03 + 3.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4 - 4i)T - 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (-10.2 + 2.75i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.30 + 1.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.93 - 7.23i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.902 + 3.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.53 - 7.53i)T - 71iT^{2} \)
73 \( 1 + (6.92 + 1.85i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 + (-4.33 + 1.16i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−11.79536090007897887458134130670, −11.09387183396756756296776757696, −9.967719528988079282051770113521, −8.946174670478772091200862249021, −8.093399596644278514558950579285, −7.32879761687602824019534571263, −5.78245223410039776567715802594, −4.64223798245322953100661734637, −4.06773188885726098886611962177, −1.76813811738880312680782878669, 2.11429463733161214632399722775, 2.85568418651825069592644585056, 3.93022304147133930090178029505, 6.20665369677389800544920284157, 6.83629692671489527011342527300, 7.991496862603880819130597879039, 8.644796579960968270905472827752, 10.48112104471594669763645781870, 10.80664369194842431446576995045, 11.99634000281006869963361206249

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