Properties

Label 2-273-91.33-c1-0-8
Degree $2$
Conductor $273$
Sign $0.867 - 0.497i$
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  + (−0.0881 + 0.328i)2-s i·3-s + (1.63 + 0.942i)4-s + (0.552 + 2.06i)5-s + (0.328 + 0.0881i)6-s + (2.31 − 1.28i)7-s + (−0.935 + 0.935i)8-s − 9-s − 0.727·10-s + (−0.152 + 0.152i)11-s + (0.942 − 1.63i)12-s + (−3.58 + 0.408i)13-s + (0.219 + 0.873i)14-s + (2.06 − 0.552i)15-s + (1.65 + 2.87i)16-s + (−2.64 + 4.58i)17-s + ⋯
L(s)  = 1  + (−0.0623 + 0.232i)2-s − 0.577i·3-s + (0.815 + 0.471i)4-s + (0.247 + 0.922i)5-s + (0.134 + 0.0359i)6-s + (0.873 − 0.486i)7-s + (−0.330 + 0.330i)8-s − 0.333·9-s − 0.230·10-s + (−0.0460 + 0.0460i)11-s + (0.271 − 0.471i)12-s + (−0.993 + 0.113i)13-s + (0.0586 + 0.233i)14-s + (0.532 − 0.142i)15-s + (0.414 + 0.718i)16-s + (−0.641 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.867 - 0.497i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50620 + 0.401226i\)
\(L(\frac12)\) \(\approx\) \(1.50620 + 0.401226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.31 + 1.28i)T \)
13 \( 1 + (3.58 - 0.408i)T \)
good2 \( 1 + (0.0881 - 0.328i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.552 - 2.06i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.152 - 0.152i)T - 11iT^{2} \)
17 \( 1 + (2.64 - 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \)
23 \( 1 + (-5.58 + 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 2.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 + 0.681i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.73 + 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.28 + 8.53i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.95 - 2.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (12.3 - 3.31i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.97 - 3.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.8 - 3.16i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 2.81iT - 61T^{2} \)
67 \( 1 + (-1.19 - 1.19i)T + 67iT^{2} \)
71 \( 1 + (-0.653 + 2.44i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.00884 + 0.0329i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.28 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.72 + 7.72i)T - 83iT^{2} \)
89 \( 1 + (-2.16 + 8.08i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.90 + 2.11i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.85684605792764525845238815918, −11.06027487656995103929126884724, −10.44951381328856685967439409234, −8.882584263621120175818581299252, −7.75894778987312503898707445418, −7.09106948135411881425108448039, −6.41442065295025396554104313492, −4.90544621579746579227042069395, −3.11625718664740241919952376163, −1.99885742438578831955221150082, 1.53775833341907410607993740972, 3.02571641325742706579234198300, 5.13306108917601289262550580335, 5.18246870842428191484824922333, 6.88865459033664846011803745064, 8.061146669268871689193973361339, 9.251394423327180920565208303799, 9.813372574978192004824993962626, 11.00930021941161805897951134559, 11.71001080788018223090231043550

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