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Properties

Label 2-273-7.4-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.110 - 0.993i$
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  + (0.698 + 1.21i)2-s + (−0.5 + 0.866i)3-s + (0.0229 − 0.0398i)4-s + (0.198 + 0.344i)5-s − 1.39·6-s + (1.69 + 2.02i)7-s + 2.85·8-s + (−0.499 − 0.866i)9-s + (−0.278 + 0.481i)10-s + (−0.824 + 1.42i)11-s + (0.0229 + 0.0398i)12-s − 13-s + (−1.26 + 3.47i)14-s − 0.397·15-s + (1.95 + 3.38i)16-s + (−2.10 + 3.64i)17-s + ⋯
L(s)  = 1  + (0.494 + 0.856i)2-s + (−0.288 + 0.499i)3-s + (0.0114 − 0.0199i)4-s + (0.0889 + 0.154i)5-s − 0.570·6-s + (0.642 + 0.766i)7-s + 1.01·8-s + (−0.166 − 0.288i)9-s + (−0.0879 + 0.152i)10-s + (−0.248 + 0.430i)11-s + (0.00663 + 0.0114i)12-s − 0.277·13-s + (−0.338 + 0.928i)14-s − 0.102·15-s + (0.488 + 0.845i)16-s + (−0.510 + 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.110 - 0.993i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12052 + 1.25148i\)
\(L(\frac12)\) \(\approx\) \(1.12052 + 1.25148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 2.02i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.698 - 1.21i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.198 - 0.344i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.824 - 1.42i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.10 - 3.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.54 + 4.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.20 - 2.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 + (-1.92 + 3.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.13 + 8.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 - 8.47T + 43T^{2} \)
47 \( 1 + (-2.02 - 3.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.66 + 8.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.827 + 1.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.50 + 9.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.221 + 0.384i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.74T + 71T^{2} \)
73 \( 1 + (3.32 - 5.76i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.28 + 3.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + (2.14 + 3.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.26T + 97T^{2} \)
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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

−12.21596025709079665403964076848, −11.04645790767311013727155214834, −10.47774576969156431708512160118, −9.211247793627702186262805234654, −8.172901666906521480973782720297, −6.99656694598115588082734939461, −6.04395042092682602455885726708, −5.11439603757945285334540161233, −4.30057521362831166887860596099, −2.23075075370641900588501971548, 1.39001501789295944266087697260, 2.84990969770666538737859644084, 4.26904392129961784579677319497, 5.23093162366485867516378607873, 6.80742063632547042520792163101, 7.65699155800726781872394036942, 8.671406683380858058267674540379, 10.30906865275939497592087972236, 10.83791437785114514653789281240, 11.79085147174881957196082357411

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