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Properties

Label 2-273-91.16-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.780 - 0.625i$
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  + 0.656·2-s + (−0.5 + 0.866i)3-s − 1.56·4-s + (−0.0109 + 0.0190i)5-s + (−0.328 + 0.568i)6-s + (−1.55 + 2.13i)7-s − 2.34·8-s + (−0.499 − 0.866i)9-s + (−0.00720 + 0.0124i)10-s + (−1.69 + 2.93i)11-s + (0.784 − 1.35i)12-s + (−2.82 + 2.23i)13-s + (−1.02 + 1.40i)14-s + (−0.0109 − 0.0190i)15-s + 1.59·16-s + 3.32·17-s + ⋯
L(s)  = 1  + 0.464·2-s + (−0.288 + 0.499i)3-s − 0.784·4-s + (−0.00490 + 0.00850i)5-s + (−0.134 + 0.232i)6-s + (−0.589 + 0.807i)7-s − 0.828·8-s + (−0.166 − 0.288i)9-s + (−0.00227 + 0.00394i)10-s + (−0.511 + 0.885i)11-s + (0.226 − 0.392i)12-s + (−0.783 + 0.621i)13-s + (−0.273 + 0.375i)14-s + (−0.00283 − 0.00490i)15-s + 0.399·16-s + 0.807·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222824 + 0.634175i\)
\(L(\frac12)\) \(\approx\) \(0.222824 + 0.634175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.55 - 2.13i)T \)
13 \( 1 + (2.82 - 2.23i)T \)
good2 \( 1 - 0.656T + 2T^{2} \)
5 \( 1 + (0.0109 - 0.0190i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.69 - 2.93i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 + (1.66 + 2.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.50T + 23T^{2} \)
29 \( 1 + (-3.52 - 6.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.13 + 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.90T + 37T^{2} \)
41 \( 1 + (-1.27 - 2.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.49 + 7.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.08 - 7.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.68 - 9.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + (5.23 + 9.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.35 - 9.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.57 - 6.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.102 - 0.176i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.42 - 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 5.45T + 89T^{2} \)
97 \( 1 + (-4.71 + 8.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.48178452560288820433213720838, −11.55199778205953846983289756169, −10.13085143746120828275717163226, −9.529199798353844574132029311272, −8.718326528680456672011993176703, −7.29878245757511741374924243053, −5.93542094870186425837940994612, −5.08688694345682459377345531797, −4.12715995303275418973553906379, −2.71134729517054572159806691491, 0.45659984131137465812151910042, 2.99586899739182728967734209103, 4.20781173765140494372735006938, 5.48129054545272729271597769536, 6.30150885266036387123673983939, 7.69589945853010677724159613854, 8.429376582841331278192318139197, 9.909675093935408826591308343624, 10.40575104852551930761679266586, 11.86094265032038440281968931947

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