Properties

Label 2-273-91.33-c1-0-13
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: $\chi_{273} (124, \cdot )$
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

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

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