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Properties

Label 2-273-91.74-c1-0-8
Degree $2$
Conductor $273$
Sign $0.610 + 0.792i$
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  − 2.43·2-s + (0.5 + 0.866i)3-s + 3.92·4-s + (−0.613 − 1.06i)5-s + (−1.21 − 2.10i)6-s + (2.20 − 1.46i)7-s − 4.68·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.58i)10-s + (−1.74 − 3.02i)11-s + (1.96 + 3.39i)12-s + (−2.87 − 2.17i)13-s + (−5.35 + 3.57i)14-s + (0.613 − 1.06i)15-s + 3.55·16-s + 4.52·17-s + ⋯
L(s)  = 1  − 1.72·2-s + (0.288 + 0.499i)3-s + 1.96·4-s + (−0.274 − 0.475i)5-s + (−0.496 − 0.860i)6-s + (0.831 − 0.554i)7-s − 1.65·8-s + (−0.166 + 0.288i)9-s + (0.472 + 0.818i)10-s + (−0.526 − 0.911i)11-s + (0.566 + 0.981i)12-s + (−0.797 − 0.603i)13-s + (−1.43 + 0.954i)14-s + (0.158 − 0.274i)15-s + 0.888·16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527085 - 0.259353i\)
\(L(\frac12)\) \(\approx\) \(0.527085 - 0.259353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 1.46i)T \)
13 \( 1 + (2.87 + 2.17i)T \)
good2 \( 1 + 2.43T + 2T^{2} \)
5 \( 1 + (0.613 + 1.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.74 + 3.02i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.52T + 17T^{2} \)
19 \( 1 + (0.677 - 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.673T + 23T^{2} \)
29 \( 1 + (-2.64 + 4.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.99 + 8.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 + (-3.61 + 6.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.48 - 7.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.58 - 4.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.803T + 59T^{2} \)
61 \( 1 + (2.32 - 4.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.06 - 1.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.52 + 4.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.04 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.90 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + (3.59 + 6.22i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.33692881129329679887929832701, −10.47664224045660020412812770618, −9.913619792174287349351268931472, −8.815075639017241304636691903478, −7.88160730689074343388096970080, −7.71972809530840792438085671436, −5.88480430956597432800915992902, −4.44915285171663175447171157560, −2.66637623860664639029813389259, −0.798210055929101775812197716720, 1.62435351962758991586416749725, 2.77177600359856111131933405575, 5.04131157984697139788552231327, 6.77991862958062977688835197685, 7.41643473585712227553572805929, 8.199414220711841711373349583479, 9.063394291554167310091869305209, 10.02937812220695921927804732544, 10.84589415252368104764162250979, 11.87776628530705989907302511574

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