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Properties

Label 2-273-91.24-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.906 + 0.421i$
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  + (1.07 − 0.286i)2-s − i·3-s + (−0.667 + 0.385i)4-s + (−3.71 − 0.994i)5-s + (−0.286 − 1.07i)6-s + (−1.35 − 2.27i)7-s + (−2.17 + 2.17i)8-s − 9-s − 4.26·10-s + (4.06 − 4.06i)11-s + (0.385 + 0.667i)12-s + (−2.56 + 2.53i)13-s + (−2.10 − 2.04i)14-s + (−0.994 + 3.71i)15-s + (−0.931 + 1.61i)16-s + (−2.31 − 4.00i)17-s + ⋯
L(s)  = 1  + (0.757 − 0.202i)2-s − 0.577i·3-s + (−0.333 + 0.192i)4-s + (−1.66 − 0.444i)5-s + (−0.117 − 0.437i)6-s + (−0.512 − 0.858i)7-s + (−0.767 + 0.767i)8-s − 0.333·9-s − 1.34·10-s + (1.22 − 1.22i)11-s + (0.111 + 0.192i)12-s + (−0.711 + 0.702i)13-s + (−0.562 − 0.546i)14-s + (−0.256 + 0.958i)15-s + (−0.232 + 0.403i)16-s + (−0.560 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.906 + 0.421i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159672 - 0.722120i\)
\(L(\frac12)\) \(\approx\) \(0.159672 - 0.722120i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 2.27i)T \)
13 \( 1 + (2.56 - 2.53i)T \)
good2 \( 1 + (-1.07 + 0.286i)T + (1.73 - i)T^{2} \)
5 \( 1 + (3.71 + 0.994i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-4.06 + 4.06i)T - 11iT^{2} \)
17 \( 1 + (2.31 + 4.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.10 + 1.10i)T - 19iT^{2} \)
23 \( 1 + (-3.73 - 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.38 + 4.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.993 - 3.70i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.653 + 2.44i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.40 + 1.18i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.65 + 1.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.11 + 7.89i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.36 - 2.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.64 + 13.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.55iT - 61T^{2} \)
67 \( 1 + (4.37 + 4.37i)T + 67iT^{2} \)
71 \( 1 + (7.06 - 1.89i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.11 + 0.298i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.33 - 7.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.07 + 1.07i)T - 83iT^{2} \)
89 \( 1 + (-8.76 + 2.34i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.63 - 6.11i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.67063744812403488640971066720, −11.23618174260502443716667658662, −9.276780523020847235794019340957, −8.591078725125649580009577483432, −7.46756698550798622031826665269, −6.66524662114008525612324108351, −4.99280882193855596902857092798, −3.98203475975269659833791238067, −3.26616724146536303392572582138, −0.45317814608190531438925918919, 3.14395259268304018980950470754, 4.08136093391349346687068194113, 4.86999554788302017994264573667, 6.26055200310060210140705286380, 7.23496621856233485148453131946, 8.571480912085346656136263343605, 9.435712191352113605116010818160, 10.43095378489504968549797432885, 11.70288522208889753144213894676, 12.28311341938968129920347491892

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