Properties

Label 2-273-91.24-c1-0-16
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: $\chi_{273} (115, \cdot )$
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

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

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