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Properties

Label 2-273-273.68-c1-0-26
Degree $2$
Conductor $273$
Sign $0.654 + 0.756i$
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  + 0.0504i·2-s + (0.667 − 1.59i)3-s + 1.99·4-s + (1.19 − 2.07i)5-s + (0.0806 + 0.0336i)6-s + (0.00656 + 2.64i)7-s + 0.201i·8-s + (−2.11 − 2.13i)9-s + (0.104 + 0.0603i)10-s + (2.85 + 1.64i)11-s + (1.33 − 3.19i)12-s + (−3.26 + 1.53i)13-s + (−0.133 + 0.000331i)14-s + (−2.51 − 3.29i)15-s + 3.98·16-s − 6.34·17-s + ⋯
L(s)  = 1  + 0.0356i·2-s + (0.385 − 0.922i)3-s + 0.998·4-s + (0.534 − 0.926i)5-s + (0.0329 + 0.0137i)6-s + (0.00248 + 0.999i)7-s + 0.0712i·8-s + (−0.703 − 0.710i)9-s + (0.0330 + 0.0190i)10-s + (0.861 + 0.497i)11-s + (0.384 − 0.921i)12-s + (−0.904 + 0.427i)13-s + (−0.0356 + 8.84e−5i)14-s + (−0.648 − 0.850i)15-s + 0.996·16-s − 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.654 + 0.756i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.654 + 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63414 - 0.746804i\)
\(L(\frac12)\) \(\approx\) \(1.63414 - 0.746804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.667 + 1.59i)T \)
7 \( 1 + (-0.00656 - 2.64i)T \)
13 \( 1 + (3.26 - 1.53i)T \)
good2 \( 1 - 0.0504iT - 2T^{2} \)
5 \( 1 + (-1.19 + 2.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.85 - 1.64i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.34T + 17T^{2} \)
19 \( 1 + (4.30 - 2.48i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.78iT - 23T^{2} \)
29 \( 1 + (-2.15 + 1.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.40 + 2.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 + (0.726 + 1.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.219 - 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.16 - 3.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.7 - 6.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.83T + 59T^{2} \)
61 \( 1 + (-2.46 + 1.42i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.719 - 1.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.14 + 2.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.79 + 2.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.32 + 4.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 5.09T + 89T^{2} \)
97 \( 1 + (-11.0 - 6.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.18642107856765530578453181920, −11.11483208675002249871132347409, −9.567016399456887668823686360588, −8.866150387160431680976970399429, −7.924148990123702974624235970428, −6.59134006857535267865382187107, −6.17415297781559636867724833181, −4.62568688928893891181901664432, −2.51245815730282425912008762456, −1.77357394147677470240486064079, 2.31691861720708044076285189334, 3.36855516231362339784409453852, 4.66320215239932239897328392659, 6.31154267555430133739137383586, 6.91597132213815368685127856160, 8.149534231596786867874648479278, 9.449180677912405721313339989666, 10.34980032859512749998061319663, 10.89101259780616561249711044036, 11.57770942369489289210492942241

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