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Properties

Label 2-273-273.230-c1-0-29
Degree $2$
Conductor $273$
Sign $0.705 + 0.708i$
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  + (0.755 + 0.436i)2-s + (1.51 − 0.832i)3-s + (−0.619 − 1.07i)4-s + 0.294·5-s + (1.51 + 0.0335i)6-s + (−1.73 − 1.99i)7-s − 2.82i·8-s + (1.61 − 2.52i)9-s + (0.222 + 0.128i)10-s + (1.54 + 0.890i)11-s + (−1.83 − 1.11i)12-s + (−2.29 + 2.78i)13-s + (−0.438 − 2.26i)14-s + (0.447 − 0.245i)15-s + (−0.00636 + 0.0110i)16-s + (3.35 + 5.81i)17-s + ⋯
L(s)  = 1  + (0.534 + 0.308i)2-s + (0.876 − 0.480i)3-s + (−0.309 − 0.536i)4-s + 0.131·5-s + (0.616 + 0.0137i)6-s + (−0.655 − 0.755i)7-s − 0.999i·8-s + (0.537 − 0.842i)9-s + (0.0704 + 0.0406i)10-s + (0.464 + 0.268i)11-s + (−0.529 − 0.321i)12-s + (−0.635 + 0.772i)13-s + (−0.117 − 0.605i)14-s + (0.115 − 0.0633i)15-s + (−0.00159 + 0.00275i)16-s + (0.813 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.705 + 0.708i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78811 - 0.743172i\)
\(L(\frac12)\) \(\approx\) \(1.78811 - 0.743172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 + 0.832i)T \)
7 \( 1 + (1.73 + 1.99i)T \)
13 \( 1 + (2.29 - 2.78i)T \)
good2 \( 1 + (-0.755 - 0.436i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 0.294T + 5T^{2} \)
11 \( 1 + (-1.54 - 0.890i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.35 - 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.08 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.31 - 3.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.98 + 1.14i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.64iT - 31T^{2} \)
37 \( 1 + (-3.69 + 6.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.49 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.03 - 3.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.42T + 47T^{2} \)
53 \( 1 - 2.91iT - 53T^{2} \)
59 \( 1 + (3.25 + 5.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.2 - 7.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 5.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.16 - 4.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.89iT - 73T^{2} \)
79 \( 1 - 6.38T + 79T^{2} \)
83 \( 1 + 9.51T + 83T^{2} \)
89 \( 1 + (-2.34 + 4.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.7 - 7.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.22625851507778727618556974900, −10.65413783784633105739973421252, −9.594358181221726731370504593152, −9.196300492438528151441069097933, −7.59385223199860564485061306034, −6.87177132756074480179179467707, −5.87296599175794488159959651563, −4.34147307640487736049446762553, −3.42604670596515507448530244986, −1.44314977877195550473726679088, 2.68332268225701497320483076374, 3.26172488228273756265931139805, 4.63858652241897174688093984054, 5.63615635945627626022798455665, 7.37092980579627794317257422786, 8.259077898863734795114511325656, 9.328045347185745717736576061324, 9.792651491620284899940725964755, 11.28784798697529505127953016047, 12.18530751974606459590664270835

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