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Properties

Label 2-273-91.5-c1-0-4
Degree $2$
Conductor $273$
Sign $0.0176 - 0.999i$
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  + (0.341 + 1.27i)2-s + (−0.866 + 0.5i)3-s + (0.223 − 0.129i)4-s + (−0.271 + 0.0726i)5-s + (−0.933 − 0.933i)6-s + (2.39 − 1.12i)7-s + (2.10 + 2.10i)8-s + (0.499 − 0.866i)9-s + (−0.185 − 0.320i)10-s + (−1.49 + 5.58i)11-s + (−0.129 + 0.223i)12-s + (2.92 + 2.10i)13-s + (2.24 + 2.67i)14-s + (0.198 − 0.198i)15-s + (−1.70 + 2.95i)16-s + (−3.11 − 5.40i)17-s + ⋯
L(s)  = 1  + (0.241 + 0.901i)2-s + (−0.499 + 0.288i)3-s + (0.111 − 0.0645i)4-s + (−0.121 + 0.0325i)5-s + (−0.380 − 0.380i)6-s + (0.905 − 0.423i)7-s + (0.745 + 0.745i)8-s + (0.166 − 0.288i)9-s + (−0.0585 − 0.101i)10-s + (−0.451 + 1.68i)11-s + (−0.0372 + 0.0645i)12-s + (0.812 + 0.583i)13-s + (0.600 + 0.714i)14-s + (0.0512 − 0.0512i)15-s + (−0.427 + 0.739i)16-s + (−0.756 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0176 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0176 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0176 - 0.999i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.0176 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04720 + 1.02885i\)
\(L(\frac12)\) \(\approx\) \(1.04720 + 1.02885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.39 + 1.12i)T \)
13 \( 1 + (-2.92 - 2.10i)T \)
good2 \( 1 + (-0.341 - 1.27i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.271 - 0.0726i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.49 - 5.58i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.11 + 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0654 - 0.0175i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.343 - 0.198i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 + (-2.04 + 7.64i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.13 + 0.572i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.99 + 2.99i)T + 41iT^{2} \)
43 \( 1 + 2.41iT - 43T^{2} \)
47 \( 1 + (-2.34 - 8.75i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.74 + 6.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.27 + 0.342i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.67 + 3.85i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.85 + 1.30i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.75 + 9.75i)T - 71iT^{2} \)
73 \( 1 + (13.2 + 3.54i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.27 + 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.12 + 4.12i)T + 83iT^{2} \)
89 \( 1 + (3.31 + 12.3i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.34 - 6.34i)T + 97iT^{2} \)
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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

−11.87214415414792289692406967250, −11.26394065642737161637514390746, −10.37790432446544507743956302422, −9.272096516909237068600212316878, −7.82111550088398147351466646030, −7.21225446937724249316072549389, −6.21087846511957298610796590340, −4.92243015926803935568951149014, −4.40760940363526337710959136420, −1.96714497151210239029789093703, 1.33746178115731106710775486532, 2.86091548668038373885092133457, 4.15655342926914319738313048927, 5.56159311856938467624271876257, 6.50403437181846028104305539213, 8.058659011430008261275955844675, 8.536326835954529792677055951253, 10.42713048820418546964171083963, 10.89647990678610510618156733647, 11.57518335294195983449141076025

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