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Properties

Label 2-273-91.62-c2-0-5
Degree $2$
Conductor $273$
Sign $-0.757 - 0.652i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.853 − 0.492i)2-s + (−1.5 + 0.866i)3-s + (−1.51 + 2.62i)4-s − 0.237·5-s + (−0.853 + 1.47i)6-s + (6.93 − 0.944i)7-s + 6.92i·8-s + (1.5 − 2.59i)9-s + (−0.202 + 0.116i)10-s + (−17.0 + 9.83i)11-s − 5.24i·12-s + (−8.73 − 9.62i)13-s + (5.45 − 4.22i)14-s + (0.355 − 0.205i)15-s + (−2.64 − 4.57i)16-s + (−2.31 − 1.33i)17-s + ⋯
L(s)  = 1  + (0.426 − 0.246i)2-s + (−0.5 + 0.288i)3-s + (−0.378 + 0.655i)4-s − 0.0474·5-s + (−0.142 + 0.246i)6-s + (0.990 − 0.134i)7-s + 0.865i·8-s + (0.166 − 0.288i)9-s + (−0.0202 + 0.0116i)10-s + (−1.54 + 0.894i)11-s − 0.437i·12-s + (−0.671 − 0.740i)13-s + (0.389 − 0.301i)14-s + (0.0237 − 0.0136i)15-s + (−0.165 − 0.286i)16-s + (−0.135 − 0.0784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.757 - 0.652i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ -0.757 - 0.652i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.297454 + 0.800519i\)
\(L(\frac12)\) \(\approx\) \(0.297454 + 0.800519i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (-6.93 + 0.944i)T \)
13 \( 1 + (8.73 + 9.62i)T \)
good2 \( 1 + (-0.853 + 0.492i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 0.237T + 25T^{2} \)
11 \( 1 + (17.0 - 9.83i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (2.31 + 1.33i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (11.8 - 20.6i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.0 - 27.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-9.98 - 17.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 51.7T + 961T^{2} \)
37 \( 1 + (-20.5 + 11.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-15.2 - 26.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-18.8 + 32.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 33.1T + 2.20e3T^{2} \)
53 \( 1 + 7.94T + 2.80e3T^{2} \)
59 \( 1 + (26.0 - 45.0i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (39.7 + 22.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.2 + 22.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-7.90 - 4.56i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 75.7T + 5.32e3T^{2} \)
79 \( 1 - 53.9T + 6.24e3T^{2} \)
83 \( 1 - 128.T + 6.88e3T^{2} \)
89 \( 1 + (-27.4 - 47.6i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (23.0 - 39.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.23070000496633190078430972012, −11.10826451424654821469576904210, −10.44025632057550309196192754645, −9.266446998893787036595138150681, −7.82830563400078194452139218655, −7.57667893093865440633880517839, −5.46591019916891806748606684136, −4.95184420315259282318982838605, −3.78913782928831160754116253700, −2.24448922880224091638533199309, 0.38360621865661501034505731336, 2.28326503029852276696902384569, 4.43058003190376344733652372140, 5.15400074438694984429793223065, 6.05940166497229163349437897788, 7.24125795849675533989060627042, 8.325426178707325360856669575928, 9.407197947333970746581222926300, 10.77141374246654753862163194148, 11.05119877057958025334238906582

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