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Properties

Label 2-273-91.59-c1-0-13
Degree $2$
Conductor $273$
Sign $0.978 + 0.208i$
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  + (0.837 + 0.837i)2-s + (0.866 − 0.5i)3-s − 0.598i·4-s + (−1.58 − 0.424i)5-s + (1.14 + 0.306i)6-s + (2.46 − 0.955i)7-s + (2.17 − 2.17i)8-s + (0.499 − 0.866i)9-s + (−0.970 − 1.68i)10-s + (−2.81 − 0.754i)11-s + (−0.299 − 0.518i)12-s + (2.68 + 2.40i)13-s + (2.86 + 1.26i)14-s + (−1.58 + 0.424i)15-s + 2.44·16-s − 0.811·17-s + ⋯
L(s)  = 1  + (0.591 + 0.591i)2-s + (0.499 − 0.288i)3-s − 0.299i·4-s + (−0.708 − 0.189i)5-s + (0.466 + 0.125i)6-s + (0.932 − 0.361i)7-s + (0.769 − 0.769i)8-s + (0.166 − 0.288i)9-s + (−0.306 − 0.531i)10-s + (−0.849 − 0.227i)11-s + (−0.0863 − 0.149i)12-s + (0.743 + 0.668i)13-s + (0.765 + 0.338i)14-s + (−0.408 + 0.109i)15-s + 0.611·16-s − 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92186 - 0.202317i\)
\(L(\frac12)\) \(\approx\) \(1.92186 - 0.202317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 + 0.955i)T \)
13 \( 1 + (-2.68 - 2.40i)T \)
good2 \( 1 + (-0.837 - 0.837i)T + 2iT^{2} \)
5 \( 1 + (1.58 + 0.424i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.81 + 0.754i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 0.811T + 17T^{2} \)
19 \( 1 + (-1.66 - 6.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 1.83iT - 23T^{2} \)
29 \( 1 + (2.42 - 4.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.01 - 3.78i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.89 - 2.89i)T - 37iT^{2} \)
41 \( 1 + (0.392 + 1.46i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.14 - 1.23i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.325 - 1.21i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.12 - 1.95i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.73 - 4.73i)T + 59iT^{2} \)
61 \( 1 + (0.446 + 0.258i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.33 + 12.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.21 - 8.27i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.87 - 2.64i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.17 + 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.44 - 7.44i)T - 83iT^{2} \)
89 \( 1 + (-5.98 - 5.98i)T + 89iT^{2} \)
97 \( 1 + (-16.5 - 4.44i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.99493935935898494925750850337, −10.95514990594787092951083888320, −10.09225179599906426172012704211, −8.616055969694366587842423515718, −7.88283069951824788995644896503, −7.01144884925126966429004060936, −5.77634256928465985585927103353, −4.65714092809759655907156226057, −3.68409500200751893883369678805, −1.53543114038400928372591079062, 2.27632405952145236686034220141, 3.38822832574335303092881571566, 4.46219550974320414255062518146, 5.41573101962113262548151309280, 7.44171530640393925711762604893, 8.001931692707759400370512427724, 8.905417599147295359042850617932, 10.37024780597729455274670624304, 11.32493288771572672397653529976, 11.67475148016413986501165965305

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