Properties

Label 2-273-91.19-c1-0-5
Degree $2$
Conductor $273$
Sign $0.989 - 0.145i$
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  + (−2.14 − 0.575i)2-s + i·3-s + (2.54 + 1.46i)4-s + (3.44 − 0.922i)5-s + (0.575 − 2.14i)6-s + (2.25 + 1.38i)7-s + (−1.47 − 1.47i)8-s − 9-s − 7.92·10-s + (−2.44 − 2.44i)11-s + (−1.46 + 2.54i)12-s + (−0.668 + 3.54i)13-s + (−4.04 − 4.26i)14-s + (0.922 + 3.44i)15-s + (−0.624 − 1.08i)16-s + (1.26 − 2.18i)17-s + ⋯
L(s)  = 1  + (−1.51 − 0.406i)2-s + 0.577i·3-s + (1.27 + 0.734i)4-s + (1.53 − 0.412i)5-s + (0.234 − 0.876i)6-s + (0.852 + 0.522i)7-s + (−0.520 − 0.520i)8-s − 0.333·9-s − 2.50·10-s + (−0.736 − 0.736i)11-s + (−0.423 + 0.734i)12-s + (−0.185 + 0.982i)13-s + (−1.08 − 1.13i)14-s + (0.238 + 0.889i)15-s + (−0.156 − 0.270i)16-s + (0.306 − 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.989 - 0.145i$
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: \((2,\ 273,\ (\ :1/2),\ 0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835429 + 0.0611772i\)
\(L(\frac12)\) \(\approx\) \(0.835429 + 0.0611772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 - 1.38i)T \)
13 \( 1 + (0.668 - 3.54i)T \)
good2 \( 1 + (2.14 + 0.575i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-3.44 + 0.922i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.44 + 2.44i)T + 11iT^{2} \)
17 \( 1 + (-1.26 + 2.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.15 - 3.15i)T + 19iT^{2} \)
23 \( 1 + (-2.64 + 1.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.12 - 1.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.54 - 5.75i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.71 + 10.1i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.06 - 0.553i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.23 + 1.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.75 - 6.55i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.54 + 4.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.20 - 8.21i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + (6.52 - 6.52i)T - 67iT^{2} \)
71 \( 1 + (12.8 + 3.43i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.73 - 1.26i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.45 - 7.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.573 - 0.573i)T + 83iT^{2} \)
89 \( 1 + (8.03 + 2.15i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.06 + 15.1i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.44528820100635438984428923703, −10.71464065963005260364924615959, −9.864803729849663947678438329721, −9.125450843750067114629211130785, −8.621535134547815451376010559191, −7.40813215469738405811921752653, −5.79149646604454200905654612668, −4.98163262251146857239219783334, −2.66840033409664014527611550316, −1.51850790119485471859492205164, 1.29689931535208063265309794970, 2.47877591054106652199791497896, 5.16021338010942728979842447767, 6.22275979514060105134201898779, 7.34798245320435004115150785998, 7.84325874203870289720875663648, 9.033614247425143758438508540980, 10.05039331158355723141693485464, 10.41817707141053494549877465530, 11.47812492696068933090208086547

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