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Properties

Label 2-273-21.20-c1-0-12
Degree $2$
Conductor $273$
Sign $0.686 - 0.727i$
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  − 0.0920i·2-s + (−0.749 + 1.56i)3-s + 1.99·4-s + 3.32·5-s + (0.143 + 0.0690i)6-s + (−0.804 + 2.52i)7-s − 0.367i·8-s + (−1.87 − 2.34i)9-s − 0.305i·10-s − 2.71i·11-s + (−1.49 + 3.10i)12-s − i·13-s + (0.232 + 0.0740i)14-s + (−2.49 + 5.18i)15-s + 3.94·16-s − 5.56·17-s + ⋯
L(s)  = 1  − 0.0650i·2-s + (−0.432 + 0.901i)3-s + 0.995·4-s + 1.48·5-s + (0.0586 + 0.0281i)6-s + (−0.304 + 0.952i)7-s − 0.129i·8-s + (−0.625 − 0.780i)9-s − 0.0967i·10-s − 0.817i·11-s + (−0.431 + 0.897i)12-s − 0.277i·13-s + (0.0620 + 0.0197i)14-s + (−0.643 + 1.33i)15-s + 0.987·16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.686 - 0.727i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.686 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46150 + 0.630094i\)
\(L(\frac12)\) \(\approx\) \(1.46150 + 0.630094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.749 - 1.56i)T \)
7 \( 1 + (0.804 - 2.52i)T \)
13 \( 1 + iT \)
good2 \( 1 + 0.0920iT - 2T^{2} \)
5 \( 1 - 3.32T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
17 \( 1 + 5.56T + 17T^{2} \)
19 \( 1 - 0.453iT - 19T^{2} \)
23 \( 1 - 6.06iT - 23T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 + 2.57iT - 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 4.72T + 43T^{2} \)
47 \( 1 - 4.34T + 47T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 + 2.38T + 59T^{2} \)
61 \( 1 + 13.7iT - 61T^{2} \)
67 \( 1 + 7.22T + 67T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 - 1.59iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 0.388T + 83T^{2} \)
89 \( 1 + 1.19T + 89T^{2} \)
97 \( 1 - 6.25iT - 97T^{2} \)
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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

−11.68519583821368891702658593256, −11.12666768760086134483541218250, −10.08176775704870061794689300960, −9.465975243594316747969610845587, −8.461055949007236601662403631067, −6.58553161715728440465386238084, −5.96572799931597906665071027220, −5.24371677947682474792020609669, −3.29147370028310115822284570762, −2.12171519210094924706336119426, 1.59830577720303043215540587115, 2.57150757362387632619855581099, 4.81334830622294593243155982481, 6.21935547353797204190246082093, 6.65120033890413616639479077785, 7.45692444885051528558653584255, 8.914631681591941278063941183112, 10.27046626066630602859140484132, 10.67826891849831699653685328501, 11.84206768937962169855312561042

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