Properties

Label 2-273-273.32-c1-0-7
Degree $2$
Conductor $273$
Sign $0.596 + 0.802i$
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  + (−1.49 − 1.49i)2-s + (−1.00 + 1.41i)3-s + 2.49i·4-s + (0.189 − 0.706i)5-s + (3.61 − 0.612i)6-s + (−2.64 + 0.127i)7-s + (0.737 − 0.737i)8-s + (−0.987 − 2.83i)9-s + (−1.34 + 0.774i)10-s + (3.26 + 0.874i)11-s + (−3.51 − 2.49i)12-s + (3.48 + 0.929i)13-s + (4.15 + 3.76i)14-s + (0.807 + 0.975i)15-s + 2.77·16-s + 4.62·17-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)2-s + (−0.579 + 0.815i)3-s + 1.24i·4-s + (0.0846 − 0.315i)5-s + (1.47 − 0.250i)6-s + (−0.998 + 0.0482i)7-s + (0.260 − 0.260i)8-s + (−0.329 − 0.944i)9-s + (−0.424 + 0.245i)10-s + (0.984 + 0.263i)11-s + (−1.01 − 0.721i)12-s + (0.966 + 0.257i)13-s + (1.10 + 1.00i)14-s + (0.208 + 0.251i)15-s + 0.693·16-s + 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.596 + 0.802i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517496 - 0.260308i\)
\(L(\frac12)\) \(\approx\) \(0.517496 - 0.260308i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.00 - 1.41i)T \)
7 \( 1 + (2.64 - 0.127i)T \)
13 \( 1 + (-3.48 - 0.929i)T \)
good2 \( 1 + (1.49 + 1.49i)T + 2iT^{2} \)
5 \( 1 + (-0.189 + 0.706i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.26 - 0.874i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 4.62T + 17T^{2} \)
19 \( 1 + (1.69 + 6.31i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.23T + 23T^{2} \)
29 \( 1 + (-0.0742 - 0.0428i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.75 + 6.53i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.78 - 3.78i)T + 37iT^{2} \)
41 \( 1 + (-11.3 + 3.04i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.89 - 4.55i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.11 - 0.297i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.966 + 0.558i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0904 + 0.0904i)T - 59iT^{2} \)
61 \( 1 + (-0.604 + 1.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.33 - 1.96i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.179 - 0.671i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.83 - 1.02i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.74 - 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.66 + 4.66i)T + 83iT^{2} \)
89 \( 1 + (8.63 - 8.63i)T - 89iT^{2} \)
97 \( 1 + (7.29 + 1.95i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.38999340911927180127728657599, −10.86442772031085669406372059246, −9.613507241989911903974345056676, −9.449450901145461785797312113948, −8.513825533053364942204815684477, −6.79039704446093312303354083947, −5.70817762433088073726820917660, −4.11855726063650048938138168266, −3.01326861732023887735939593272, −0.929537771476377443791623371600, 1.08618145191779942542608293062, 3.45592569926075756020883213865, 5.77072929399928430595424712495, 6.28258737490604709489746450584, 7.06343364835785509363654558170, 8.052729353771191189258400937319, 8.927037160501292988052617064816, 10.03411239105139068777677395249, 10.81692772203594846644047238334, 12.16912045472024968842769073230

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