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Properties

Label 2-273-91.69-c2-0-8
Degree $2$
Conductor $273$
Sign $0.989 + 0.146i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.42 − 1.97i)2-s + (−1.5 − 0.866i)3-s + (5.84 + 10.1i)4-s + 3.37·5-s + (3.42 + 5.93i)6-s + (−6.27 − 3.10i)7-s − 30.4i·8-s + (1.5 + 2.59i)9-s + (−11.5 − 6.68i)10-s + (3.88 + 2.24i)11-s − 20.2i·12-s + (−12.4 + 3.83i)13-s + (15.3 + 23.0i)14-s + (−5.06 − 2.92i)15-s + (−36.8 + 63.8i)16-s + (11.4 − 6.60i)17-s + ⋯
L(s)  = 1  + (−1.71 − 0.989i)2-s + (−0.5 − 0.288i)3-s + (1.46 + 2.52i)4-s + 0.675·5-s + (0.571 + 0.989i)6-s + (−0.896 − 0.443i)7-s − 3.80i·8-s + (0.166 + 0.288i)9-s + (−1.15 − 0.668i)10-s + (0.353 + 0.203i)11-s − 1.68i·12-s + (−0.955 + 0.295i)13-s + (1.09 + 1.64i)14-s + (−0.337 − 0.194i)15-s + (−2.30 + 3.98i)16-s + (0.672 − 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ 0.989 + 0.146i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.494438 - 0.0363665i\)
\(L(\frac12)\) \(\approx\) \(0.494438 - 0.0363665i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (6.27 + 3.10i)T \)
13 \( 1 + (12.4 - 3.83i)T \)
good2 \( 1 + (3.42 + 1.97i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 3.37T + 25T^{2} \)
11 \( 1 + (-3.88 - 2.24i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + (-11.4 + 6.60i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.4 - 19.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (12.8 - 22.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (2.16 - 3.75i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 31.3T + 961T^{2} \)
37 \( 1 + (-1.37 - 0.794i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-34.2 + 59.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-34.5 - 59.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 3.52T + 2.20e3T^{2} \)
53 \( 1 - 10.3T + 2.80e3T^{2} \)
59 \( 1 + (-50.3 - 87.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-19.3 + 11.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-66.7 - 38.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (58.6 - 33.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 73.9T + 5.32e3T^{2} \)
79 \( 1 - 43.0T + 6.24e3T^{2} \)
83 \( 1 + 6.83T + 6.88e3T^{2} \)
89 \( 1 + (-53.3 + 92.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-18.9 - 32.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.62601326929713941999254216350, −10.32635779850523893484941836521, −9.850185917335657048957068980296, −9.286176768699795857536899857599, −7.78476020832416314959577425302, −7.16045520106245697406995700623, −6.02371818339693252607688080747, −3.78407500742713715881943896000, −2.39955731982246086724862348245, −1.06235279674184846801637763315, 0.56531741587176951242454951897, 2.42970548162338938981104812335, 5.18298445200214614021030196566, 6.06920321197413461478752681827, 6.72293932891232205397688410191, 7.87549511475209982084734211320, 9.010413362049665856850185142445, 9.785241965283457284579266451016, 10.13657302500351221311954803394, 11.29705976320649396448510222502

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