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Properties

Label 2-273-7.2-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.975 + 0.220i$
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  + (−0.939 + 1.62i)2-s + (0.5 + 0.866i)3-s + (−0.766 − 1.32i)4-s + (−1.43 + 2.49i)5-s − 1.87·6-s + (2.47 + 0.943i)7-s − 0.879·8-s + (−0.499 + 0.866i)9-s + (−2.70 − 4.68i)10-s + (0.326 + 0.565i)11-s + (0.766 − 1.32i)12-s + 13-s + (−3.85 + 3.13i)14-s − 2.87·15-s + (2.35 − 4.08i)16-s + (−2.26 − 3.92i)17-s + ⋯
L(s)  = 1  + (−0.664 + 1.15i)2-s + (0.288 + 0.499i)3-s + (−0.383 − 0.663i)4-s + (−0.643 + 1.11i)5-s − 0.767·6-s + (0.934 + 0.356i)7-s − 0.310·8-s + (−0.166 + 0.288i)9-s + (−0.855 − 1.48i)10-s + (0.0983 + 0.170i)11-s + (0.221 − 0.383i)12-s + 0.277·13-s + (−1.03 + 0.838i)14-s − 0.743·15-s + (0.589 − 1.02i)16-s + (−0.549 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.975 + 0.220i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0949531 - 0.850698i\)
\(L(\frac12)\) \(\approx\) \(0.0949531 - 0.850698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.47 - 0.943i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.939 - 1.62i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.43 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.326 - 0.565i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.26 + 3.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.37 - 4.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0714 - 0.123i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.26T + 29T^{2} \)
31 \( 1 + (2.59 + 4.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.266 - 0.460i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.63T + 41T^{2} \)
43 \( 1 + 2.83T + 43T^{2} \)
47 \( 1 + (0.305 - 0.528i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.673 - 1.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.56 - 13.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.71 - 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.631 - 1.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + (-5.53 - 9.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.65 + 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 + (-0.737 + 1.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.04T + 97T^{2} \)
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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

−12.10676825405749496766622679386, −11.31060635366774960586881463810, −10.40787727311512830709049559179, −9.261312044145206351270219328380, −8.357330487603973626162675660299, −7.63640906052443234992544790590, −6.77675720093792783302817855309, −5.62092558154097685886872637320, −4.19465953983801381491283976564, −2.71962472121342457220243379205, 0.819986593895102864130326887990, 2.02929172386769140725004579519, 3.73439487558590327048987781051, 4.90238070902066351556779660012, 6.57353749757084942768590017156, 8.155581062260980435069647393648, 8.468284554875407702042510971129, 9.376688383173481569453604378498, 10.77427331499573775528692201467, 11.24963699523758831141668179103

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