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Properties

Label 2-273-91.30-c1-0-5
Degree $2$
Conductor $273$
Sign $0.980 - 0.194i$
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  + (1.27 − 0.733i)2-s − 3-s + (0.0756 − 0.131i)4-s + (1.88 + 1.08i)5-s + (−1.27 + 0.733i)6-s + (0.427 + 2.61i)7-s + 2.71i·8-s + 9-s + 3.18·10-s − 5.49i·11-s + (−0.0756 + 0.131i)12-s + (3.04 + 1.93i)13-s + (2.45 + 3.00i)14-s + (−1.88 − 1.08i)15-s + (2.13 + 3.70i)16-s + (−1.99 + 3.45i)17-s + ⋯
L(s)  = 1  + (0.898 − 0.518i)2-s − 0.577·3-s + (0.0378 − 0.0655i)4-s + (0.841 + 0.485i)5-s + (−0.518 + 0.299i)6-s + (0.161 + 0.986i)7-s + 0.958i·8-s + 0.333·9-s + 1.00·10-s − 1.65i·11-s + (−0.0218 + 0.0378i)12-s + (0.843 + 0.536i)13-s + (0.656 + 0.802i)14-s + (−0.485 − 0.280i)15-s + (0.534 + 0.926i)16-s + (−0.483 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.980 - 0.194i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.980 - 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82333 + 0.178923i\)
\(L(\frac12)\) \(\approx\) \(1.82333 + 0.178923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (-0.427 - 2.61i)T \)
13 \( 1 + (-3.04 - 1.93i)T \)
good2 \( 1 + (-1.27 + 0.733i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.88 - 1.08i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.49iT - 11T^{2} \)
17 \( 1 + (1.99 - 3.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 4.33iT - 19T^{2} \)
23 \( 1 + (0.862 + 1.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.36 + 7.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.95 - 1.70i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.86 - 4.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.74 - 3.89i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.33 - 1.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.08 + 7.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.27 + 0.738i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 + 1.10iT - 67T^{2} \)
71 \( 1 + (-6.60 + 3.81i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.82 - 3.93i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.56 + 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + (-3.17 + 1.83i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.57 - 2.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.81854862938456749790708752831, −11.24991228412930650478765857212, −10.49920485095271140182683196434, −8.989261372141766822497022074144, −8.315806480235487727285176282562, −6.26689414626365091391523256787, −5.96685894281526596246232313880, −4.74447195254224727127641355162, −3.35468346381852358089520655990, −2.14612120767910098998055469570, 1.42903529725585989891036794191, 3.89603203563815088880674023806, 4.87762294518785781596032174888, 5.62126931393262939521943899916, 6.73319931629653659988103391675, 7.50621233514579236251842860591, 9.250098844137523381426349675987, 10.06207322713159950173296421632, 10.80874693471117867530618287899, 12.34849410326073375131865851287

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