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Properties

Label 2-273-13.3-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.765 - 0.643i$
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  + (1.16 + 2.01i)2-s + (0.5 + 0.866i)3-s + (−1.71 + 2.96i)4-s + 0.900·5-s + (−1.16 + 2.01i)6-s + (−0.5 + 0.866i)7-s − 3.33·8-s + (−0.499 + 0.866i)9-s + (1.04 + 1.81i)10-s + (−1.45 − 2.51i)11-s − 3.42·12-s + (1.54 − 3.25i)13-s − 2.33·14-s + (0.450 + 0.780i)15-s + (−0.450 − 0.780i)16-s + (−0.834 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.823 + 1.42i)2-s + (0.288 + 0.499i)3-s + (−0.857 + 1.48i)4-s + 0.402·5-s + (−0.475 + 0.823i)6-s + (−0.188 + 0.327i)7-s − 1.17·8-s + (−0.166 + 0.288i)9-s + (0.331 + 0.574i)10-s + (−0.437 − 0.757i)11-s − 0.989·12-s + (0.429 − 0.902i)13-s − 0.622·14-s + (0.116 + 0.201i)15-s + (−0.112 − 0.195i)16-s + (−0.202 + 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.765 - 0.643i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.765 - 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706328 + 1.93612i\)
\(L(\frac12)\) \(\approx\) \(0.706328 + 1.93612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-1.54 + 3.25i)T \)
good2 \( 1 + (-1.16 - 2.01i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.900T + 5T^{2} \)
11 \( 1 + (1.45 + 2.51i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.834 - 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 3.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.66 + 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + (4.30 + 7.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.66 - 6.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.21 + 3.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.95T + 47T^{2} \)
53 \( 1 + 0.141T + 53T^{2} \)
59 \( 1 + (4.74 - 8.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.25 + 2.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.00491 + 0.00850i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.04 - 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.57T + 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 - 1.58T + 83T^{2} \)
89 \( 1 + (4.16 + 7.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.85 + 6.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.79900814107522297416730687237, −11.38257289397822602784374725597, −10.27096005819456032017834814521, −9.083871025024127941368598143377, −8.219873626589894361224857467381, −7.31013607990565586779113782013, −5.89425244752649667088649730763, −5.55350360634906955875292657616, −4.20251468133369929335861778013, −2.98410815948756040506829103718, 1.52486228965156254413938250320, 2.65582446043950492361987166498, 3.92633861587325962106111808087, 5.02169251533633956181670026240, 6.36622261477000242572130915250, 7.56572146221832715137896669198, 9.038783891113047961656514243106, 9.944420580938999487452460297519, 10.72385497460370232787119132581, 11.79161196913066876871785831432

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