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Properties

Label 2-273-91.20-c1-0-8
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: $\chi_{273} (202, \cdot )$
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

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

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