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Properties

Label 2-273-91.23-c1-0-8
Degree $2$
Conductor $273$
Sign $0.980 - 0.195i$
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  + 0.915i·2-s + (0.5 − 0.866i)3-s + 1.16·4-s + (−2.26 − 1.30i)5-s + (0.792 + 0.457i)6-s + (2.57 + 0.608i)7-s + 2.89i·8-s + (−0.499 − 0.866i)9-s + (1.19 − 2.07i)10-s + (2.20 + 1.27i)11-s + (0.581 − 1.00i)12-s + (0.523 − 3.56i)13-s + (−0.557 + 2.35i)14-s + (−2.26 + 1.30i)15-s − 0.325·16-s + 6.19·17-s + ⋯
L(s)  = 1  + 0.647i·2-s + (0.288 − 0.499i)3-s + 0.581·4-s + (−1.01 − 0.585i)5-s + (0.323 + 0.186i)6-s + (0.973 + 0.230i)7-s + 1.02i·8-s + (−0.166 − 0.288i)9-s + (0.378 − 0.656i)10-s + (0.664 + 0.383i)11-s + (0.167 − 0.290i)12-s + (0.145 − 0.989i)13-s + (−0.148 + 0.629i)14-s + (−0.585 + 0.337i)15-s − 0.0813·16-s + 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55859 + 0.153532i\)
\(L(\frac12)\) \(\approx\) \(1.55859 + 0.153532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.57 - 0.608i)T \)
13 \( 1 + (-0.523 + 3.56i)T \)
good2 \( 1 - 0.915iT - 2T^{2} \)
5 \( 1 + (2.26 + 1.30i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.20 - 1.27i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.19T + 17T^{2} \)
19 \( 1 + (-2.36 + 1.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.89T + 23T^{2} \)
29 \( 1 + (1.77 + 3.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.91 - 4.56i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.68iT - 37T^{2} \)
41 \( 1 + (3.98 - 2.30i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.11 - 3.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.53 + 0.887i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.42 + 5.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.25iT - 59T^{2} \)
61 \( 1 + (6.18 + 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.536 - 0.309i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.48 - 3.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.8 - 6.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.621 + 1.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.10iT - 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + (1.32 + 0.767i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.89542485430465458999362802785, −11.40552297972545088067529771731, −9.993475134319351085021078507041, −8.453756318147344020378767495100, −7.963762954395169602736209647815, −7.34846044710861584225225335799, −5.97939620807102840891208288721, −4.95004069150170231447230815378, −3.40189063635528015235764934766, −1.58513769932380620795960385806, 1.77838662123423708571736291237, 3.51706300326752007841202511958, 4.00813167066585372883127006850, 5.78615303802889695375366839797, 7.27928137324140850852459053581, 7.82753623830523620790348483711, 9.175859886264470724594630605253, 10.26707124851208395539015920378, 11.09387944238182210771579547483, 11.68461509793670915159159482246

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