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Properties

Label 2-273-21.5-c1-0-6
Degree $2$
Conductor $273$
Sign $0.657 - 0.753i$
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  + (−1.15 − 0.665i)2-s + (−0.732 + 1.56i)3-s + (−0.114 − 0.198i)4-s + (−0.0955 + 0.165i)5-s + (1.88 − 1.32i)6-s + (2.46 − 0.955i)7-s + 2.96i·8-s + (−1.92 − 2.30i)9-s + (0.220 − 0.127i)10-s + (−2.17 + 1.25i)11-s + (0.394 − 0.0342i)12-s + i·13-s + (−3.47 − 0.540i)14-s + (−0.189 − 0.271i)15-s + (1.74 − 3.02i)16-s + (3.30 + 5.71i)17-s + ⋯
L(s)  = 1  + (−0.815 − 0.470i)2-s + (−0.423 + 0.906i)3-s + (−0.0571 − 0.0990i)4-s + (−0.0427 + 0.0739i)5-s + (0.771 − 0.539i)6-s + (0.932 − 0.361i)7-s + 1.04i·8-s + (−0.641 − 0.766i)9-s + (0.0696 − 0.0401i)10-s + (−0.657 + 0.379i)11-s + (0.113 − 0.00989i)12-s + 0.277i·13-s + (−0.929 − 0.144i)14-s + (−0.0489 − 0.0700i)15-s + (0.436 − 0.755i)16-s + (0.800 + 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.609489 + 0.276907i\)
\(L(\frac12)\) \(\approx\) \(0.609489 + 0.276907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.732 - 1.56i)T \)
7 \( 1 + (-2.46 + 0.955i)T \)
13 \( 1 - iT \)
good2 \( 1 + (1.15 + 0.665i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.0955 - 0.165i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.17 - 1.25i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.30 - 5.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.52 - 3.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.428 - 0.247i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.66iT - 29T^{2} \)
31 \( 1 + (7.59 - 4.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.433 + 0.750i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.55T + 41T^{2} \)
43 \( 1 + 2.13T + 43T^{2} \)
47 \( 1 + (-4.61 + 7.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.02 - 2.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.58 + 9.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.06 - 0.615i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.38 - 4.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (-6.07 + 3.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-3.24 + 5.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.0iT - 97T^{2} \)
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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

−11.58438752802836150297747837490, −10.78473039617047793774996625590, −10.31162822727907651517370197664, −9.394727103296958062876801049624, −8.446911830222964569798745457436, −7.45994105962610362506681705986, −5.62922182438434052485891077575, −4.98559240519014215424056580032, −3.53685613241423838948190322485, −1.51084753205677450076253231308, 0.795059551200506788858846458539, 2.79975069586382690764914325131, 4.89014559353807278866056679280, 5.87762391842726019325597122586, 7.36257964087502338423243877433, 7.68430604418288836042404485796, 8.642833977787557992250011504666, 9.636142462219532003506137224676, 10.95472652017764868664985568178, 11.75607457165086798270184085381

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