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Properties

Label 2-273-91.54-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.966 + 0.254i$
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  + (1.92 − 1.92i)2-s + (−0.866 − 0.5i)3-s − 5.43i·4-s + (−3.41 + 0.914i)5-s + (−2.63 + 0.705i)6-s + (2.45 − 0.996i)7-s + (−6.62 − 6.62i)8-s + (0.499 + 0.866i)9-s + (−4.82 + 8.34i)10-s + (−0.426 + 0.114i)11-s + (−2.71 + 4.70i)12-s + (3.60 − 0.0874i)13-s + (2.80 − 6.64i)14-s + (3.41 + 0.914i)15-s − 14.6·16-s + 1.43·17-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (−0.499 − 0.288i)3-s − 2.71i·4-s + (−1.52 + 0.409i)5-s + (−1.07 + 0.288i)6-s + (0.926 − 0.376i)7-s + (−2.34 − 2.34i)8-s + (0.166 + 0.288i)9-s + (−1.52 + 2.64i)10-s + (−0.128 + 0.0344i)11-s + (−0.784 + 1.35i)12-s + (0.999 − 0.0242i)13-s + (0.749 − 1.77i)14-s + (0.881 + 0.236i)15-s − 3.67·16-s + 0.347·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.966 + 0.254i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.966 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.229376 - 1.76949i\)
\(L(\frac12)\) \(\approx\) \(0.229376 - 1.76949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.45 + 0.996i)T \)
13 \( 1 + (-3.60 + 0.0874i)T \)
good2 \( 1 + (-1.92 + 1.92i)T - 2iT^{2} \)
5 \( 1 + (3.41 - 0.914i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.426 - 0.114i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 + (-0.340 + 1.27i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.18iT - 23T^{2} \)
29 \( 1 + (-3.82 - 6.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.187 + 0.698i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.719 - 0.719i)T + 37iT^{2} \)
41 \( 1 + (0.748 - 2.79i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.20 - 4.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.242 - 0.906i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.48 - 4.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.29 - 2.29i)T - 59iT^{2} \)
61 \( 1 + (11.2 - 6.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.64 - 6.13i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.487 - 1.82i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (12.2 + 3.29i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.77 - 3.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.33 - 2.33i)T + 83iT^{2} \)
89 \( 1 + (-7.39 + 7.39i)T - 89iT^{2} \)
97 \( 1 + (2.51 - 0.673i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.58954569671142875737612597361, −10.86052309199918727592932623083, −10.48869806323832191991950985198, −8.611524097431671846812220816078, −7.30912811654777251135281481341, −6.12713649340788587856236305858, −4.76990798228678526218123607181, −4.13339357800909826761268269186, −2.93493715755951316650882898232, −1.07847714348026333613154922127, 3.52026354758306321358262241782, 4.29712926523573460250141112773, 5.20119376937712676083865333800, 6.09662096845828705710896499812, 7.48532553511184100392435071095, 7.997555881178811916367679879085, 8.879900190117279386832954358122, 11.07273709502804774799468598563, 11.81804197288320851846533169428, 12.23901257033902132357803898001

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