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Properties

Label 2-273-91.88-c1-0-7
Degree $2$
Conductor $273$
Sign $-0.769 + 0.638i$
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  + (−1.77 − 1.02i)2-s − 3-s + (1.09 + 1.90i)4-s + (−0.341 + 0.197i)5-s + (1.77 + 1.02i)6-s + (1.69 − 2.02i)7-s − 0.400i·8-s + 9-s + 0.807·10-s − 1.56i·11-s + (−1.09 − 1.90i)12-s + (1.46 + 3.29i)13-s + (−5.09 + 1.85i)14-s + (0.341 − 0.197i)15-s + (1.78 − 3.09i)16-s + (−3.41 − 5.91i)17-s + ⋯
L(s)  = 1  + (−1.25 − 0.724i)2-s − 0.577·3-s + (0.548 + 0.950i)4-s + (−0.152 + 0.0881i)5-s + (0.724 + 0.418i)6-s + (0.641 − 0.766i)7-s − 0.141i·8-s + 0.333·9-s + 0.255·10-s − 0.472i·11-s + (−0.316 − 0.548i)12-s + (0.406 + 0.913i)13-s + (−1.36 + 0.497i)14-s + (0.0881 − 0.0508i)15-s + (0.446 − 0.773i)16-s + (−0.827 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148426 - 0.411135i\)
\(L(\frac12)\) \(\approx\) \(0.148426 - 0.411135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (-1.69 + 2.02i)T \)
13 \( 1 + (-1.46 - 3.29i)T \)
good2 \( 1 + (1.77 + 1.02i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.341 - 0.197i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
17 \( 1 + (3.41 + 5.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 7.79iT - 19T^{2} \)
23 \( 1 + (2.39 - 4.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.94 + 6.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.14 + 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.01 - 0.586i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.17 + 3.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.65 + 6.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.92 + 2.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.964 - 1.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.91 + 5.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 2.79iT - 67T^{2} \)
71 \( 1 + (-7.66 - 4.42i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.23 - 1.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.55 - 2.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.01iT - 83T^{2} \)
89 \( 1 + (-10.8 - 6.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.58 + 5.53i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27769557928273720628388925149, −10.89828247854020310196916505406, −9.522050296658631688999974046480, −9.047462663362678564346935213028, −7.67963032885555342664496551058, −7.01329940369178418170615172038, −5.37164488472918068720458444671, −4.04984040701559132830015538316, −2.17782152790713838560426537682, −0.55629499991966394383610513861, 1.66357252077397472914012692613, 4.08683643711642996123885532610, 5.68331111863939046713044082051, 6.33743085724817836578092936364, 7.75693425728691974623448303387, 8.264835865496117692726414931517, 9.215864610727852798509677695355, 10.40576185432328733526133515244, 10.87487109319356105987902305028, 12.37105342586257590173226253011

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