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Properties

Label 2-273-91.80-c1-0-0
Degree $2$
Conductor $273$
Sign $0.795 - 0.605i$
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  + (−0.639 − 2.38i)2-s − i·3-s + (−3.54 + 2.04i)4-s + (−0.746 + 2.78i)5-s + (−2.38 + 0.639i)6-s + (−2.52 + 0.794i)7-s + (3.65 + 3.65i)8-s − 9-s + 7.11·10-s + (0.990 + 0.990i)11-s + (2.04 + 3.54i)12-s + (−3.49 + 0.872i)13-s + (3.50 + 5.51i)14-s + (2.78 + 0.746i)15-s + (2.29 − 3.96i)16-s + (3.77 + 6.54i)17-s + ⋯
L(s)  = 1  + (−0.451 − 1.68i)2-s − 0.577i·3-s + (−1.77 + 1.02i)4-s + (−0.333 + 1.24i)5-s + (−0.973 + 0.260i)6-s + (−0.953 + 0.300i)7-s + (1.29 + 1.29i)8-s − 0.333·9-s + 2.25·10-s + (0.298 + 0.298i)11-s + (0.591 + 1.02i)12-s + (−0.970 + 0.242i)13-s + (0.937 + 1.47i)14-s + (0.719 + 0.192i)15-s + (0.572 − 0.992i)16-s + (0.916 + 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.795 - 0.605i$
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: \((2,\ 273,\ (\ :1/2),\ 0.795 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.265893 + 0.0896677i\)
\(L(\frac12)\) \(\approx\) \(0.265893 + 0.0896677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 - 0.794i)T \)
13 \( 1 + (3.49 - 0.872i)T \)
good2 \( 1 + (0.639 + 2.38i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.746 - 2.78i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.990 - 0.990i)T + 11iT^{2} \)
17 \( 1 + (-3.77 - 6.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.88 + 4.88i)T + 19iT^{2} \)
23 \( 1 + (1.97 + 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.75 + 6.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.19 - 1.39i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.20 - 0.591i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.38 - 8.90i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.04 - 1.75i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.833 + 0.223i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.886 - 1.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.19 - 1.39i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 4.18iT - 61T^{2} \)
67 \( 1 + (3.93 - 3.93i)T - 67iT^{2} \)
71 \( 1 + (1.75 + 6.54i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.17 - 8.10i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.411 - 0.713i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.15 + 2.15i)T + 83iT^{2} \)
89 \( 1 + (0.666 + 2.48i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.79 - 2.35i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.92721533020202950623831902580, −11.05904463557487057282055643299, −10.26352186242717269780407053343, −9.540248908729507651125470619513, −8.408463791717121155694176491506, −7.20435134190701965790652031216, −6.19252753618387871633625524350, −4.08584720624295515359098948707, −3.02947256678184424229801700168, −2.09754812270105010344903113751, 0.23970096101114486740474210357, 3.78255938851167297264053075828, 5.02455587838148619433565834088, 5.69808539423042925130348995351, 7.02227202147824116016900884955, 7.86315983963681810094556215579, 8.931931094911472402481641682040, 9.431224854041561217193049105819, 10.33129415395517145935907550750, 12.05852335354360781422561198362

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