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Properties

Label 2-273-273.44-c1-0-10
Degree $2$
Conductor $273$
Sign $0.301 - 0.953i$
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  + (−2.69 + 0.722i)2-s + (1.57 + 0.716i)3-s + (5.01 − 2.89i)4-s + (1.51 − 0.407i)5-s + (−4.76 − 0.792i)6-s + (−0.780 + 2.52i)7-s + (−7.47 + 7.47i)8-s + (1.97 + 2.26i)9-s + (−3.80 + 2.19i)10-s + (0.998 + 0.267i)11-s + (9.97 − 0.971i)12-s + (−0.226 − 3.59i)13-s + (0.279 − 7.37i)14-s + (2.68 + 0.447i)15-s + (8.95 − 15.5i)16-s + (2.22 + 3.85i)17-s + ⋯
L(s)  = 1  + (−1.90 + 0.510i)2-s + (0.910 + 0.413i)3-s + (2.50 − 1.44i)4-s + (0.679 − 0.182i)5-s + (−1.94 − 0.323i)6-s + (−0.295 + 0.955i)7-s + (−2.64 + 2.64i)8-s + (0.657 + 0.753i)9-s + (−1.20 + 0.694i)10-s + (0.300 + 0.0806i)11-s + (2.87 − 0.280i)12-s + (−0.0628 − 0.998i)13-s + (0.0745 − 1.97i)14-s + (0.694 + 0.115i)15-s + (2.23 − 3.87i)16-s + (0.540 + 0.936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.301 - 0.953i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674381 + 0.494258i\)
\(L(\frac12)\) \(\approx\) \(0.674381 + 0.494258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.57 - 0.716i)T \)
7 \( 1 + (0.780 - 2.52i)T \)
13 \( 1 + (0.226 + 3.59i)T \)
good2 \( 1 + (2.69 - 0.722i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.51 + 0.407i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.998 - 0.267i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.22 - 3.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.324 - 1.21i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.753 + 1.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.07iT - 29T^{2} \)
31 \( 1 + (-3.60 - 0.964i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.748 + 0.200i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.06 + 2.06i)T + 41iT^{2} \)
43 \( 1 - 9.19iT - 43T^{2} \)
47 \( 1 + (0.858 + 3.20i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.565 - 0.326i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.19 - 0.589i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.158 - 0.275i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 + 0.446i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.88 - 2.88i)T + 71iT^{2} \)
73 \( 1 + (-0.832 + 3.10i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.46 + 9.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.87 + 3.87i)T + 83iT^{2} \)
89 \( 1 + (4.28 + 15.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.36 + 6.36i)T - 97iT^{2} \)
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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

−11.75711252697056151185060044507, −10.45032250092083598585392976826, −9.894971660697800232586210807006, −9.214373667062347661257413610052, −8.375228174988947257654512097914, −7.76169663603191459941664343835, −6.34663383616390776013400309603, −5.49943304772994303992138108132, −2.91385031205871569284152037117, −1.74183107360921397667866486713, 1.19436777486235399845430599351, 2.45063253247870439172338884405, 3.64096693675627167243538476574, 6.56193466786527870556736639346, 7.09469053401796870238955089381, 8.006786844311215272918606039565, 9.115424463896615151064025443400, 9.620583937061390041411136189757, 10.34987105541580660948891614287, 11.45952163807490017256897958208

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