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Properties

Label 2-273-91.88-c1-0-4
Degree $2$
Conductor $273$
Sign $0.990 - 0.140i$
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  + (−1.33 − 0.771i)2-s + 3-s + (0.189 + 0.328i)4-s + (−1.27 + 0.733i)5-s + (−1.33 − 0.771i)6-s + (1.52 + 2.16i)7-s + 2.50i·8-s + 9-s + 2.26·10-s + 2.23i·11-s + (0.189 + 0.328i)12-s + (3.57 − 0.453i)13-s + (−0.369 − 4.06i)14-s + (−1.27 + 0.733i)15-s + (2.30 − 3.99i)16-s + (1.26 + 2.18i)17-s + ⋯
L(s)  = 1  + (−0.944 − 0.545i)2-s + 0.577·3-s + (0.0947 + 0.164i)4-s + (−0.568 + 0.328i)5-s + (−0.545 − 0.314i)6-s + (0.576 + 0.817i)7-s + 0.883i·8-s + 0.333·9-s + 0.715·10-s + 0.672i·11-s + (0.0547 + 0.0947i)12-s + (0.992 − 0.125i)13-s + (−0.0986 − 1.08i)14-s + (−0.328 + 0.189i)15-s + (0.576 − 0.999i)16-s + (0.305 + 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886655 + 0.0623878i\)
\(L(\frac12)\) \(\approx\) \(0.886655 + 0.0623878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (-1.52 - 2.16i)T \)
13 \( 1 + (-3.57 + 0.453i)T \)
good2 \( 1 + (1.33 + 0.771i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.27 - 0.733i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
17 \( 1 + (-1.26 - 2.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 0.957iT - 19T^{2} \)
23 \( 1 + (-1.32 + 2.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.728 + 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.89 - 1.66i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.41 - 3.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.52 + 2.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 + 5.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.05 - 5.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.74 + 3.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.664 - 0.383i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 9.64iT - 67T^{2} \)
71 \( 1 + (2.50 + 1.44i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.3 + 6.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.88 + 3.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.89iT - 83T^{2} \)
89 \( 1 + (-8.78 - 5.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.44 + 3.14i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.60857543625858555271728191481, −10.93549106543401008629344019755, −9.965950488449849091511811307244, −8.999809326368481757770950373634, −8.320921720893951952775762942920, −7.52405132281818190963554322766, −5.94149107111213863288618230533, −4.52984272719844554313580018279, −2.95534103595130951857821925594, −1.62695909484382230094088645446, 1.01936439693956429146022561566, 3.46432823267248861817218624860, 4.42675930620955679538455165655, 6.22284878575765376745524676568, 7.49565472193613043186272596917, 8.004628888763477451382056882475, 8.770796744839246386877843002639, 9.686101444658730732983321348248, 10.77626169628335196482429509564, 11.70064468779616885113158441759

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