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Properties

Label 2-273-91.80-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.993 - 0.109i$
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  + (0.568 + 2.12i)2-s + i·3-s + (−2.44 + 1.40i)4-s + (−0.248 + 0.926i)5-s + (−2.12 + 0.568i)6-s + (1.82 + 1.91i)7-s + (−1.27 − 1.27i)8-s − 9-s − 2.10·10-s + (−2.00 − 2.00i)11-s + (−1.40 − 2.44i)12-s + (2.97 − 2.03i)13-s + (−3.03 + 4.95i)14-s + (−0.926 − 0.248i)15-s + (−0.844 + 1.46i)16-s + (−2.53 − 4.39i)17-s + ⋯
L(s)  = 1  + (0.401 + 1.49i)2-s + 0.577i·3-s + (−1.22 + 0.704i)4-s + (−0.111 + 0.414i)5-s + (−0.865 + 0.231i)6-s + (0.688 + 0.724i)7-s + (−0.449 − 0.449i)8-s − 0.333·9-s − 0.665·10-s + (−0.603 − 0.603i)11-s + (−0.406 − 0.704i)12-s + (0.825 − 0.564i)13-s + (−0.810 + 1.32i)14-s + (−0.239 − 0.0640i)15-s + (−0.211 + 0.365i)16-s + (−0.615 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.993 - 0.109i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.993 - 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0831908 + 1.51207i\)
\(L(\frac12)\) \(\approx\) \(0.0831908 + 1.51207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-1.82 - 1.91i)T \)
13 \( 1 + (-2.97 + 2.03i)T \)
good2 \( 1 + (-0.568 - 2.12i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.248 - 0.926i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.00 + 2.00i)T + 11iT^{2} \)
17 \( 1 + (2.53 + 4.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.314 - 0.314i)T + 19iT^{2} \)
23 \( 1 + (2.92 + 1.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.91 - 8.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.21 + 0.861i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.09 + 1.36i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.13 + 4.21i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.57 + 1.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.10 - 2.43i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.30 - 7.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.83 + 0.758i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + (7.12 - 7.12i)T - 67iT^{2} \)
71 \( 1 + (0.206 + 0.770i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.01 + 3.79i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.44 - 4.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.86 + 2.86i)T + 83iT^{2} \)
89 \( 1 + (2.70 + 10.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.64 + 1.24i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.56523027523850028914202178938, −11.23948540509463356050574473662, −10.61962786833875079061213803481, −9.003302080531442690317285409512, −8.395898045504293320059901023138, −7.43562396886595917799514033202, −6.24089822590320216101853309050, −5.40272594582486899690431258715, −4.54766291607786820236177829622, −2.95582952215421805045929949250, 1.18152169344059024671135424648, 2.35681017685857510751144586084, 4.01592682534251115471572687147, 4.72626187436752849040246522115, 6.35196430184323103422709572908, 7.72801419583189300827811005023, 8.637341901672308330244674247081, 9.974709836575315629946245499532, 10.72987108766464480469254210211, 11.56825158670437402572992512298

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