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Properties

Label 2-273-91.76-c1-0-12
Degree $2$
Conductor $273$
Sign $0.965 - 0.261i$
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  + (−0.189 + 0.707i)2-s + (0.866 − 0.5i)3-s + (1.26 + 0.731i)4-s + (1.23 − 1.23i)5-s + (0.189 + 0.707i)6-s + (2.32 − 1.25i)7-s + (−1.79 + 1.79i)8-s + (0.499 − 0.866i)9-s + (0.639 + 1.10i)10-s + (−3.18 − 0.854i)11-s + 1.46·12-s + (−2.53 − 2.56i)13-s + (0.449 + 1.88i)14-s + (0.451 − 1.68i)15-s + (0.532 + 0.923i)16-s + (0.433 − 0.751i)17-s + ⋯
L(s)  = 1  + (−0.134 + 0.500i)2-s + (0.499 − 0.288i)3-s + (0.633 + 0.365i)4-s + (0.551 − 0.551i)5-s + (0.0774 + 0.289i)6-s + (0.879 − 0.475i)7-s + (−0.634 + 0.634i)8-s + (0.166 − 0.288i)9-s + (0.202 + 0.350i)10-s + (−0.961 − 0.257i)11-s + 0.422·12-s + (−0.701 − 0.712i)13-s + (0.120 + 0.504i)14-s + (0.116 − 0.434i)15-s + (0.133 + 0.230i)16-s + (0.105 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.965 - 0.261i$
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: \((2,\ 273,\ (\ :1/2),\ 0.965 - 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70751 + 0.227657i\)
\(L(\frac12)\) \(\approx\) \(1.70751 + 0.227657i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.32 + 1.25i)T \)
13 \( 1 + (2.53 + 2.56i)T \)
good2 \( 1 + (0.189 - 0.707i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.23 + 1.23i)T - 5iT^{2} \)
11 \( 1 + (3.18 + 0.854i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.433 + 0.751i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.01 - 3.80i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.77 - 2.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.220 - 0.220i)T - 31iT^{2} \)
37 \( 1 + (3.57 + 0.957i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.90 + 0.509i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.99 - 5.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.68 + 3.68i)T + 47iT^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + (8.89 - 2.38i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.78 + 2.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.01 + 3.80i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.9 + 3.20i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.55 + 5.55i)T + 73iT^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (-3.80 + 3.80i)T - 83iT^{2} \)
89 \( 1 + (-3.26 + 12.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.756 + 2.82i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.11972935100143102150422220282, −10.95755418801018623812245093618, −10.02148736064452246199064780599, −8.755250715846869225479712055321, −7.83025449426552530764381530175, −7.45018499040551104272773120541, −5.93918478440180178364972555848, −5.00818184713178593442781550539, −3.18708836979459055319315738314, −1.80264436883970850215192963757, 2.08337236005203263559003025439, 2.70655541709438848199443986519, 4.58487398877297958072470946225, 5.80266015638587544338704504478, 6.98024057330044916507484792242, 8.040440285504220756769860596576, 9.277153478655328187146639438547, 10.11087956226609083972945587800, 10.77969983879266616213901267021, 11.70922714732113754395608931186

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