Properties

Label 2-273-91.41-c1-0-5
Degree $2$
Conductor $273$
Sign $0.996 + 0.0841i$
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  + (−2.16 − 0.581i)2-s + (0.866 − 0.5i)3-s + (2.63 + 1.52i)4-s + (1.87 + 1.87i)5-s + (−2.16 + 0.581i)6-s + (1.25 − 2.32i)7-s + (−1.65 − 1.65i)8-s + (0.499 − 0.866i)9-s + (−2.98 − 5.16i)10-s + (−1.20 + 4.51i)11-s + 3.04·12-s + (1.52 + 3.26i)13-s + (−4.07 + 4.31i)14-s + (2.56 + 0.687i)15-s + (−0.417 − 0.723i)16-s + (−2.18 + 3.78i)17-s + ⋯
L(s)  = 1  + (−1.53 − 0.410i)2-s + (0.499 − 0.288i)3-s + (1.31 + 0.760i)4-s + (0.839 + 0.839i)5-s + (−0.885 + 0.237i)6-s + (0.475 − 0.879i)7-s + (−0.584 − 0.584i)8-s + (0.166 − 0.288i)9-s + (−0.942 − 1.63i)10-s + (−0.364 + 1.36i)11-s + 0.877·12-s + (0.423 + 0.906i)13-s + (−1.09 + 1.15i)14-s + (0.662 + 0.177i)15-s + (−0.104 − 0.180i)16-s + (−0.530 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.996 + 0.0841i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.996 + 0.0841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866434 - 0.0365337i\)
\(L(\frac12)\) \(\approx\) \(0.866434 - 0.0365337i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 2.32i)T \)
13 \( 1 + (-1.52 - 3.26i)T \)
good2 \( 1 + (2.16 + 0.581i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.87 - 1.87i)T + 5iT^{2} \)
11 \( 1 + (1.20 - 4.51i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.194 + 0.0521i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.20 + 4.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.20 + 9.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.75 - 6.75i)T + 31iT^{2} \)
37 \( 1 + (-1.22 + 4.58i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.136 - 0.508i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.49 - 1.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.928 - 0.928i)T - 47iT^{2} \)
53 \( 1 - 1.95T + 53T^{2} \)
59 \( 1 + (1.76 + 6.57i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.40 + 0.377i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.44 + 5.37i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.75 - 8.75i)T - 73iT^{2} \)
79 \( 1 + 4.46T + 79T^{2} \)
83 \( 1 + (-5.42 - 5.42i)T + 83iT^{2} \)
89 \( 1 + (15.0 + 4.03i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.84 + 0.763i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.39759988227378307110375191318, −10.61304398263879157880182292151, −10.01427211851701542169634827980, −9.168176859927339215136292028914, −8.152790721260515602832331380319, −7.18462336218940482586961809236, −6.56908358239967665391670081092, −4.42202488480642841489968056171, −2.52935072547926357067610288972, −1.61266895838772305389064909568, 1.23005121708570364087962570452, 2.85318761577837893490943824715, 5.13093668440547657221190885560, 5.93425626975558350335442375945, 7.46513117892722873286232911755, 8.500919342793761704949953911359, 8.910312295263071284675014188327, 9.562085938902407673933716277190, 10.71057522010272522393627965118, 11.45179534093786610762255327710

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