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Properties

Label 2-273-91.33-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.891 + 0.452i$
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  + (−0.604 + 2.25i)2-s − i·3-s + (−2.99 − 1.73i)4-s + (0.721 + 2.69i)5-s + (2.25 + 0.604i)6-s + (−1.83 − 1.90i)7-s + (2.41 − 2.41i)8-s − 9-s − 6.51·10-s + (−4.28 + 4.28i)11-s + (−1.73 + 2.99i)12-s + (−0.711 + 3.53i)13-s + (5.40 − 3.00i)14-s + (2.69 − 0.721i)15-s + (0.530 + 0.918i)16-s + (0.641 − 1.11i)17-s + ⋯
L(s)  = 1  + (−0.427 + 1.59i)2-s − 0.577i·3-s + (−1.49 − 0.865i)4-s + (0.322 + 1.20i)5-s + (0.921 + 0.246i)6-s + (−0.695 − 0.718i)7-s + (0.854 − 0.854i)8-s − 0.333·9-s − 2.06·10-s + (−1.29 + 1.29i)11-s + (−0.499 + 0.865i)12-s + (−0.197 + 0.980i)13-s + (1.44 − 0.802i)14-s + (0.695 − 0.186i)15-s + (0.132 + 0.229i)16-s + (0.155 − 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133158 - 0.557245i\)
\(L(\frac12)\) \(\approx\) \(0.133158 - 0.557245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (1.83 + 1.90i)T \)
13 \( 1 + (0.711 - 3.53i)T \)
good2 \( 1 + (0.604 - 2.25i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.721 - 2.69i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.28 - 4.28i)T - 11iT^{2} \)
17 \( 1 + (-0.641 + 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.37 - 1.37i)T - 19iT^{2} \)
23 \( 1 + (1.85 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.92 + 0.515i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-11.4 - 3.05i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.19 - 4.44i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.94 - 1.70i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.99 - 2.67i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.30 - 7.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.2 + 3.00i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 3.67iT - 61T^{2} \)
67 \( 1 + (-1.20 - 1.20i)T + 67iT^{2} \)
71 \( 1 + (1.73 - 6.47i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.17 - 4.37i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.418 + 0.724i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.01 - 2.01i)T - 83iT^{2} \)
89 \( 1 + (-1.29 + 4.81i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.27 + 1.68i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.83281517274039707459351618697, −11.38891731785806316404747228191, −10.01908275609502043398876061339, −9.712880921704480912417938736847, −8.073996996740716923090367453280, −7.34526615757876841250572157443, −6.75409004573035862183757425744, −6.01409202321532321573799338981, −4.55323460125652085922214000379, −2.56823083621673590539659014173, 0.48147091177073341260676966073, 2.50533957942568093595313599113, 3.47466871513913480626434002637, 4.99987554756706627132438403208, 5.83673343298242675238802115431, 8.293470669251144229703382084466, 8.712781406649170207548054985050, 9.684497375172100248436536090126, 10.38092658472718325160473122250, 11.20931059084976496330445099300

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