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Properties

Label 2-273-7.2-c1-0-11
Degree $2$
Conductor $273$
Sign $0.605 + 0.795i$
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  + (0.277 − 0.480i)2-s + (−0.5 − 0.866i)3-s + (0.846 + 1.46i)4-s + (2.02 − 3.50i)5-s − 0.554·6-s + (0.167 + 2.64i)7-s + 2.04·8-s + (−0.499 + 0.866i)9-s + (−1.12 − 1.94i)10-s + (−0.733 − 1.27i)11-s + (0.846 − 1.46i)12-s + 13-s + (1.31 + 0.652i)14-s − 4.04·15-s + (−1.12 + 1.94i)16-s + (−1.59 − 2.75i)17-s + ⋯
L(s)  = 1  + (0.196 − 0.339i)2-s + (−0.288 − 0.499i)3-s + (0.423 + 0.732i)4-s + (0.905 − 1.56i)5-s − 0.226·6-s + (0.0633 + 0.997i)7-s + 0.724·8-s + (−0.166 + 0.288i)9-s + (−0.355 − 0.615i)10-s + (−0.221 − 0.383i)11-s + (0.244 − 0.423i)12-s + 0.277·13-s + (0.351 + 0.174i)14-s − 1.04·15-s + (−0.280 + 0.486i)16-s + (−0.386 − 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45215 - 0.719871i\)
\(L(\frac12)\) \(\approx\) \(1.45215 - 0.719871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.167 - 2.64i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.277 + 0.480i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.02 + 3.50i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.733 + 1.27i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.59 + 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.05 + 3.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.84 - 3.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.44T + 29T^{2} \)
31 \( 1 + (-3.27 - 5.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.852 + 1.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.44T + 41T^{2} \)
43 \( 1 + 6.58T + 43T^{2} \)
47 \( 1 + (6.20 - 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.11 - 5.39i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.91 + 5.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.25 - 2.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.89 - 8.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.43T + 71T^{2} \)
73 \( 1 + (-8.26 - 14.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.16 - 3.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.246T + 83T^{2} \)
89 \( 1 + (-9.00 + 15.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−11.94236654386204420634075017493, −11.24390025921184481268439052877, −9.744188961437632901127297762590, −8.758100685663862475928345387155, −8.138274737051167708086182955815, −6.69272268888352113740838509064, −5.54270704294195388654929690718, −4.73986499999875799180422786415, −2.80924813478028018490616705725, −1.54787569596872038122281958164, 1.99548387673502740195641983995, 3.62107018167495888365213502459, 5.07792280714526445768236715020, 6.32073650149331713442571159701, 6.65872783778518545931359677554, 7.86031560914980767675291672222, 9.809568606214966787070653762267, 10.24168132435814981170583955194, 10.75826493427429338098711919603, 11.67191456777836840443413389740

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