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Properties

Label 2-273-91.16-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.681 - 0.732i$
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  + 1.22·2-s + (−0.5 + 0.866i)3-s − 0.495·4-s + (−2.10 + 3.64i)5-s + (−0.613 + 1.06i)6-s + (0.113 − 2.64i)7-s − 3.06·8-s + (−0.499 − 0.866i)9-s + (−2.58 + 4.47i)10-s + (−2.76 + 4.78i)11-s + (0.247 − 0.429i)12-s + (3.59 + 0.226i)13-s + (0.139 − 3.24i)14-s + (−2.10 − 3.64i)15-s − 2.76·16-s + 0.178·17-s + ⋯
L(s)  = 1  + 0.867·2-s + (−0.288 + 0.499i)3-s − 0.247·4-s + (−0.942 + 1.63i)5-s + (−0.250 + 0.433i)6-s + (0.0429 − 0.999i)7-s − 1.08·8-s + (−0.166 − 0.288i)9-s + (−0.817 + 1.41i)10-s + (−0.832 + 1.44i)11-s + (0.0715 − 0.123i)12-s + (0.998 + 0.0627i)13-s + (0.0372 − 0.866i)14-s + (−0.543 − 0.942i)15-s − 0.690·16-s + 0.0432·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.681 - 0.732i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.681 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401224 + 0.921516i\)
\(L(\frac12)\) \(\approx\) \(0.401224 + 0.921516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.113 + 2.64i)T \)
13 \( 1 + (-3.59 - 0.226i)T \)
good2 \( 1 - 1.22T + 2T^{2} \)
5 \( 1 + (2.10 - 3.64i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.76 - 4.78i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.178T + 17T^{2} \)
19 \( 1 + (-2.25 - 3.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.08T + 23T^{2} \)
29 \( 1 + (0.0731 + 0.126i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.19 - 7.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.31T + 37T^{2} \)
41 \( 1 + (-0.782 - 1.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.636 + 1.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.93 + 6.82i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.02T + 59T^{2} \)
61 \( 1 + (-1.88 - 3.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.153 + 0.266i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.62 - 2.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.53 + 6.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.30 - 3.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.85T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (-1.17 + 2.03i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.19487459870205033323161848510, −11.32680057214402630830023729212, −10.45645085919194686176778716834, −9.865364433050718383670402895971, −8.133590515163275815071010497951, −7.18006129863384016616058378673, −6.25713348110692034374855361474, −4.78420643686589427627026667792, −3.90661716997192800273517062761, −3.09625339548384863701380082600, 0.62459898540681522773622354915, 3.10671596039499944798966441802, 4.44990352810482684766959572576, 5.41361941359113147245046322763, 5.98080937901944323942000404505, 7.946307509968965445521096342155, 8.577994055820922200218468403081, 9.225249762526204497309921035969, 11.28039591181876460621312255317, 11.72489916208898826634834008241

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