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Properties

Label 2-273-91.80-c1-0-13
Degree $2$
Conductor $273$
Sign $0.292 + 0.956i$
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  + (−0.185 − 0.693i)2-s + i·3-s + (1.28 − 0.741i)4-s + (0.295 − 1.10i)5-s + (0.693 − 0.185i)6-s + (−0.650 − 2.56i)7-s + (−1.76 − 1.76i)8-s − 9-s − 0.818·10-s + (−1.26 − 1.26i)11-s + (0.741 + 1.28i)12-s + (3.60 + 0.189i)13-s + (−1.65 + 0.928i)14-s + (1.10 + 0.295i)15-s + (0.584 − 1.01i)16-s + (1.84 + 3.19i)17-s + ⋯
L(s)  = 1  + (−0.131 − 0.490i)2-s + 0.577i·3-s + (0.642 − 0.370i)4-s + (0.131 − 0.492i)5-s + (0.283 − 0.0759i)6-s + (−0.245 − 0.969i)7-s + (−0.625 − 0.625i)8-s − 0.333·9-s − 0.258·10-s + (−0.381 − 0.381i)11-s + (0.214 + 0.370i)12-s + (0.998 + 0.0525i)13-s + (−0.443 + 0.248i)14-s + (0.284 + 0.0761i)15-s + (0.146 − 0.253i)16-s + (0.447 + 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07790 - 0.797415i\)
\(L(\frac12)\) \(\approx\) \(1.07790 - 0.797415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.650 + 2.56i)T \)
13 \( 1 + (-3.60 - 0.189i)T \)
good2 \( 1 + (0.185 + 0.693i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.295 + 1.10i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.26 + 1.26i)T + 11iT^{2} \)
17 \( 1 + (-1.84 - 3.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.439 + 0.439i)T + 19iT^{2} \)
23 \( 1 + (1.96 + 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0743 + 0.128i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.64 + 1.24i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.33 - 1.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.12 - 4.19i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.81 - 3.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.291 + 0.0779i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.7 - 3.68i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.90iT - 61T^{2} \)
67 \( 1 + (8.53 - 8.53i)T - 67iT^{2} \)
71 \( 1 + (3.67 + 13.6i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.05 - 11.3i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.537 + 0.930i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.90 + 3.90i)T + 83iT^{2} \)
89 \( 1 + (-2.99 - 11.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.98 - 1.33i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.42755061081281059278005314946, −10.65139039063727706776020014074, −10.13244551724879638611320884542, −9.063667735227283919382589193791, −7.966886168319505424104343526762, −6.59275504241109400906047792203, −5.68039334446493268208129126775, −4.21255181893280983935577818527, −3.07518290684097304793618237858, −1.18339853431145674414335189055, 2.20789022817692179609874099996, 3.24762051809016545702593635062, 5.41421832976408007061326332672, 6.31355351633346900068134116748, 7.09311711129503876103368870571, 8.121969602996287491168726336864, 8.915141219575802357067347007967, 10.26699454352348636704012403834, 11.35455043734239547451233110779, 12.08591597425992256335012122543

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