Properties

Label 2-273-273.2-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.711 - 0.702i$
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  + (0.340 + 0.340i)2-s + (−1.34 − 1.09i)3-s − 1.76i·4-s + (−3.37 + 0.904i)5-s + (−0.0826 − 0.829i)6-s + (−0.304 + 2.62i)7-s + (1.28 − 1.28i)8-s + (0.591 + 2.94i)9-s + (−1.45 − 0.841i)10-s + (0.752 + 2.80i)11-s + (−1.94 + 2.36i)12-s + (−3.15 + 1.75i)13-s + (−0.998 + 0.791i)14-s + (5.51 + 2.49i)15-s − 2.66·16-s + 1.23·17-s + ⋯
L(s)  = 1  + (0.240 + 0.240i)2-s + (−0.773 − 0.633i)3-s − 0.884i·4-s + (−1.50 + 0.404i)5-s + (−0.0337 − 0.338i)6-s + (−0.115 + 0.993i)7-s + (0.453 − 0.453i)8-s + (0.197 + 0.980i)9-s + (−0.460 − 0.266i)10-s + (0.226 + 0.846i)11-s + (−0.560 + 0.684i)12-s + (−0.873 + 0.486i)13-s + (−0.266 + 0.211i)14-s + (1.42 + 0.643i)15-s − 0.665·16-s + 0.299·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0897735 + 0.218579i\)
\(L(\frac12)\) \(\approx\) \(0.0897735 + 0.218579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 + 1.09i)T \)
7 \( 1 + (0.304 - 2.62i)T \)
13 \( 1 + (3.15 - 1.75i)T \)
good2 \( 1 + (-0.340 - 0.340i)T + 2iT^{2} \)
5 \( 1 + (3.37 - 0.904i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.752 - 2.80i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
19 \( 1 + (4.44 + 1.19i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.05T + 23T^{2} \)
29 \( 1 + (1.21 - 0.698i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.15 + 0.576i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.70 - 1.70i)T + 37iT^{2} \)
41 \( 1 + (-0.777 + 2.90i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.87 + 5.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.95 - 7.31i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (10.1 - 5.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.19 + 7.19i)T - 59iT^{2} \)
61 \( 1 + (-3.19 - 5.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 9.69i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-15.4 + 4.13i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.944 - 3.52i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.31 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.80 + 1.80i)T + 83iT^{2} \)
89 \( 1 + (8.20 - 8.20i)T - 89iT^{2} \)
97 \( 1 + (-2.44 - 9.13i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.13429491718191015932854895732, −11.52588386081285534525290905355, −10.61382139687149302054893117688, −9.521318032666008711997752957822, −8.099773871953947828916469767710, −7.11238761585341295148882538566, −6.42548576144914771127713233496, −5.19709224843221144980300609322, −4.25446315761765134549058923259, −2.12876206612369129110513563642, 0.17759993495615936125298328463, 3.48091146976526241909137579814, 4.03046069028443182706425702393, 4.98375698605757835493370117756, 6.66772297693615287925951919884, 7.76650484627486229327979673926, 8.418364431535072020967895076635, 9.873836779420318736401528948357, 10.98755593065750284969868588345, 11.50397909356717260376867937153

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