Properties

Label 2-273-91.33-c1-0-16
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: $\chi_{273} (124, \cdot )$
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

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

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