Properties

Label 2-273-91.76-c1-0-3
Degree $2$
Conductor $273$
Sign $0.642 - 0.766i$
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  + (0.0473 − 0.176i)2-s + (0.866 − 0.5i)3-s + (1.70 + 0.983i)4-s + (−2.80 + 2.80i)5-s + (−0.0473 − 0.176i)6-s + (1.70 + 2.02i)7-s + (0.513 − 0.513i)8-s + (0.499 − 0.866i)9-s + (0.362 + 0.627i)10-s + (−2.53 − 0.679i)11-s + 1.96·12-s + (−1.37 + 3.33i)13-s + (0.437 − 0.205i)14-s + (−1.02 + 3.82i)15-s + (1.90 + 3.29i)16-s + (1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.0334 − 0.124i)2-s + (0.499 − 0.288i)3-s + (0.851 + 0.491i)4-s + (−1.25 + 1.25i)5-s + (−0.0193 − 0.0721i)6-s + (0.645 + 0.764i)7-s + (0.181 − 0.181i)8-s + (0.166 − 0.288i)9-s + (0.114 + 0.198i)10-s + (−0.764 − 0.204i)11-s + 0.567·12-s + (−0.380 + 0.924i)13-s + (0.117 − 0.0550i)14-s + (−0.264 + 0.987i)15-s + (0.475 + 0.822i)16-s + (0.347 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38346 + 0.645824i\)
\(L(\frac12)\) \(\approx\) \(1.38346 + 0.645824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.70 - 2.02i)T \)
13 \( 1 + (1.37 - 3.33i)T \)
good2 \( 1 + (-0.0473 + 0.176i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (2.80 - 2.80i)T - 5iT^{2} \)
11 \( 1 + (2.53 + 0.679i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.759 + 2.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-7.27 + 4.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.66 - 2.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.75 + 6.75i)T - 31iT^{2} \)
37 \( 1 + (6.77 + 1.81i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.79 - 0.747i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.43 + 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.85 - 4.85i)T + 47iT^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + (-0.00666 + 0.00178i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.65 + 3.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.10 + 7.84i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (14.5 - 3.91i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.321 + 0.321i)T + 73iT^{2} \)
79 \( 1 - 0.280T + 79T^{2} \)
83 \( 1 + (-2.42 + 2.42i)T - 83iT^{2} \)
89 \( 1 + (0.0536 - 0.200i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.197 + 0.736i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.84618292705183707581948229948, −11.27552559909997553223473441853, −10.53736371948398211233561620967, −8.898772516511760928251467816102, −7.912734572711292966849139976948, −7.28394325004098756779027601389, −6.48324774157765388196642568594, −4.58573867433473192601746479448, −3.08061636087528941527474414927, −2.48230146333342206766419637493, 1.24594006811624680498311415849, 3.25722980957878214179057514875, 4.62994639309854519967271883522, 5.36473546601977310779265519189, 7.25153770418778867188390608042, 7.87482834460996523026385515739, 8.582531403698049715222504549393, 10.10061260223848409059075536829, 10.74455728374045875601457772094, 11.79499822752320411321848868751

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