Properties

Label 2-273-91.19-c1-0-11
Degree $2$
Conductor $273$
Sign $0.621 - 0.783i$
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  + (2.18 + 0.584i)2-s + i·3-s + (2.68 + 1.55i)4-s + (2.27 − 0.609i)5-s + (−0.584 + 2.18i)6-s + (−2.33 + 1.23i)7-s + (1.76 + 1.76i)8-s − 9-s + 5.31·10-s + (−3.57 − 3.57i)11-s + (−1.55 + 2.68i)12-s + (1.62 − 3.21i)13-s + (−5.82 + 1.32i)14-s + (0.609 + 2.27i)15-s + (−0.288 − 0.499i)16-s + (−2.64 + 4.57i)17-s + ⋯
L(s)  = 1  + (1.54 + 0.413i)2-s + 0.577i·3-s + (1.34 + 0.775i)4-s + (1.01 − 0.272i)5-s + (−0.238 + 0.890i)6-s + (−0.884 + 0.466i)7-s + (0.623 + 0.623i)8-s − 0.333·9-s + 1.68·10-s + (−1.07 − 1.07i)11-s + (−0.447 + 0.775i)12-s + (0.450 − 0.892i)13-s + (−1.55 + 0.354i)14-s + (0.157 + 0.587i)15-s + (−0.0720 − 0.124i)16-s + (−0.641 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.621 - 0.783i$
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: \((2,\ 273,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53465 + 1.22547i\)
\(L(\frac12)\) \(\approx\) \(2.53465 + 1.22547i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 - 1.23i)T \)
13 \( 1 + (-1.62 + 3.21i)T \)
good2 \( 1 + (-2.18 - 0.584i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-2.27 + 0.609i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.57 + 3.57i)T + 11iT^{2} \)
17 \( 1 + (2.64 - 4.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.98 - 2.98i)T + 19iT^{2} \)
23 \( 1 + (-3.44 + 1.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.565 + 0.978i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.40 - 5.23i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.37 + 5.13i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.61 + 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.65 - 5.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.95 - 7.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.538 + 0.931i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.18 + 8.16i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.15iT - 61T^{2} \)
67 \( 1 + (3.75 - 3.75i)T - 67iT^{2} \)
71 \( 1 + (-12.9 - 3.47i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.56 - 0.418i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.31 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.21 + 7.21i)T + 83iT^{2} \)
89 \( 1 + (-8.63 - 2.31i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.32 - 16.1i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.68062889113719054540840243307, −11.16775428539238389695215988401, −10.29970223101209102331881619032, −9.234909194488602506474190104405, −8.111689264194541000553874598716, −6.40754973748218994973445451459, −5.75282001613185991628156410710, −5.18693184180762160399982522449, −3.62266438710655072236820363509, −2.75220782593228752821539166483, 2.11384599571134563045976809090, 3.02655671656123084612511081989, 4.56688973765890030320485601035, 5.56391074621036155078681062922, 6.63139903655999276621341409842, 7.23063006043270680921769842680, 9.194842529091210276539382657314, 10.08079906509143946878495333302, 11.15681352070112563816825869979, 12.01300852316978040707411990225

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