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Properties

Label 2-273-273.272-c1-0-21
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  + 0.305·2-s + (−1.47 + 0.913i)3-s − 1.90·4-s + 2.05i·5-s + (−0.450 + 0.279i)6-s + (0.946 − 2.47i)7-s − 1.19·8-s + (1.33 − 2.68i)9-s + 0.628i·10-s − 5.03·11-s + (2.80 − 1.74i)12-s + (−1.75 − 3.14i)13-s + (0.289 − 0.755i)14-s + (−1.87 − 3.02i)15-s + 3.44·16-s − 3.24·17-s + ⋯
L(s)  = 1  + 0.216·2-s + (−0.849 + 0.527i)3-s − 0.953·4-s + 0.918i·5-s + (−0.183 + 0.114i)6-s + (0.357 − 0.933i)7-s − 0.422·8-s + (0.444 − 0.895i)9-s + 0.198i·10-s − 1.51·11-s + (0.810 − 0.502i)12-s + (−0.487 − 0.872i)13-s + (0.0774 − 0.201i)14-s + (−0.484 − 0.780i)15-s + 0.861·16-s − 0.787·17-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: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0332485 - 0.0912594i\)
\(L(\frac12)\) \(\approx\) \(0.0332485 - 0.0912594i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.47 - 0.913i)T \)
7 \( 1 + (-0.946 + 2.47i)T \)
13 \( 1 + (1.75 + 3.14i)T \)
good2 \( 1 - 0.305T + 2T^{2} \)
5 \( 1 - 2.05iT - 5T^{2} \)
11 \( 1 + 5.03T + 11T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 + 4.21T + 19T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 + 5.11T + 31T^{2} \)
37 \( 1 - 4.85iT - 37T^{2} \)
41 \( 1 - 3.35iT - 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 7.14iT - 47T^{2} \)
53 \( 1 + 3.08iT - 53T^{2} \)
59 \( 1 + 12.5iT - 59T^{2} \)
61 \( 1 - 3.48iT - 61T^{2} \)
67 \( 1 - 7.77iT - 67T^{2} \)
71 \( 1 + 6.61T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 2.15T + 79T^{2} \)
83 \( 1 + 0.936iT - 83T^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−11.22906158133481350116508742447, −10.39247524268988450416905453879, −10.23345645080441577903000047179, −8.673791287140377472300934747117, −7.53547524740477952547007025595, −6.41586211109905608208142257635, −5.14195155301902389337214329886, −4.45852562835362510500840375455, −3.09753833724811128826926954006, −0.07498264032321597326554378161, 2.10258152378777008338708603117, 4.44189102801898835857099503884, 5.16839235929702763907350351663, 5.88094733103287737940892339200, 7.48083161739578832973050740920, 8.500399991142397412017572207300, 9.241294130033446890765495900472, 10.47516984274536646493916989346, 11.60367743137775341418163584722, 12.41151694627880367912140367907

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