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Properties

Label 2-273-13.3-c1-0-0
Degree $2$
Conductor $273$
Sign $0.604 - 0.796i$
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  + (−1.06 − 1.84i)2-s + (0.5 + 0.866i)3-s + (−1.26 + 2.19i)4-s − 2.65·5-s + (1.06 − 1.84i)6-s + (−0.5 + 0.866i)7-s + 1.12·8-s + (−0.499 + 0.866i)9-s + (2.82 + 4.90i)10-s + (0.329 + 0.570i)11-s − 2.53·12-s + (3.32 + 1.38i)13-s + 2.12·14-s + (−1.32 − 2.30i)15-s + (1.32 + 2.30i)16-s + (−3.06 + 5.30i)17-s + ⋯
L(s)  = 1  + (−0.752 − 1.30i)2-s + (0.288 + 0.499i)3-s + (−0.632 + 1.09i)4-s − 1.18·5-s + (0.434 − 0.752i)6-s + (−0.188 + 0.327i)7-s + 0.398·8-s + (−0.166 + 0.288i)9-s + (0.894 + 1.54i)10-s + (0.0992 + 0.171i)11-s − 0.730·12-s + (0.923 + 0.383i)13-s + 0.568·14-s + (−0.343 − 0.594i)15-s + (0.332 + 0.575i)16-s + (−0.743 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.604 - 0.796i$
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: \((2,\ 273,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405392 + 0.201206i\)
\(L(\frac12)\) \(\approx\) \(0.405392 + 0.201206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.32 - 1.38i)T \)
good2 \( 1 + (1.06 + 1.84i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.65T + 5T^{2} \)
11 \( 1 + (-0.329 - 0.570i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.06 - 5.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.62 - 4.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.76 - 3.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.435 + 0.754i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.91T + 31T^{2} \)
37 \( 1 + (0.731 + 1.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.25 + 9.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.76 + 3.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.71T + 47T^{2} \)
53 \( 1 + 1.93T + 53T^{2} \)
59 \( 1 + (6.95 - 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.09 - 7.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.69 + 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.13 - 3.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.46T + 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (-4.75 - 8.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.48 - 2.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.87952127995983483591134280666, −10.85818581810843096591331742110, −10.49116825668575014350159035667, −9.079053894470446483468658636898, −8.690663724464318165453067216093, −7.64585214059810242525701007436, −6.00572879362743356790748196060, −4.04149677556930015792589105553, −3.57169598762409832541743699862, −1.90052012833032333917315447055, 0.42576099935636519415210507762, 3.18939263938219306776796529479, 4.71928688702739114886362319259, 6.28361021389968667566908547506, 7.05277565938583489183993655572, 7.78121452883166467938703240597, 8.632313705342387270878765576005, 9.288292616803164858726511897293, 10.83601408853147403875203143366, 11.64937126687858681405847346862

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