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Properties

Label 2-273-273.149-c1-0-23
Degree $2$
Conductor $273$
Sign $0.997 - 0.0771i$
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  + 1.73·3-s + (1.73 + i)4-s + (−0.866 − 2.5i)7-s + 2.99·9-s + (2.99 + 1.73i)12-s + (−3.46 − i)13-s + (1.99 + 3.46i)16-s + (−3.83 + 3.83i)19-s + (−1.49 − 4.33i)21-s + (−4.33 + 2.5i)25-s + 5.19·27-s + (1.00 − 5.19i)28-s + (−1.13 − 0.303i)31-s + (5.19 + 2.99i)36-s + (1.06 − 3.96i)37-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.866 + 0.5i)4-s + (−0.327 − 0.944i)7-s + 0.999·9-s + (0.866 + 0.499i)12-s + (−0.960 − 0.277i)13-s + (0.499 + 0.866i)16-s + (−0.878 + 0.878i)19-s + (−0.327 − 0.944i)21-s + (−0.866 + 0.5i)25-s + 1.00·27-s + (0.188 − 0.981i)28-s + (−0.203 − 0.0545i)31-s + (0.866 + 0.499i)36-s + (0.174 − 0.651i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.997 - 0.0771i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.997 - 0.0771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93117 + 0.0745802i\)
\(L(\frac12)\) \(\approx\) \(1.93117 + 0.0745802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
7 \( 1 + (0.866 + 2.5i)T \)
13 \( 1 + (3.46 + i)T \)
good2 \( 1 + (-1.73 - i)T^{2} \)
5 \( 1 + (4.33 - 2.5i)T^{2} \)
11 \( 1 - 11iT^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.83 - 3.83i)T - 19iT^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.13 + 0.303i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.06 + 3.96i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9 + 5.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 + (11.5 - 11.5i)T - 67iT^{2} \)
71 \( 1 + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.36 + 16.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.06 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.42 - 2.52i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.13075721811619682096140581616, −10.76215945503836603191764840782, −10.11760405476867941365372753429, −9.000823340604805562211047877385, −7.67429731731366436108446736691, −7.43749487051906280962738954880, −6.17161410218688959431182686228, −4.26228655679691115691827820720, −3.29239212165727272561433624926, −2.01087686788100944455718525059, 2.07529505677522911963470350920, 2.89508173716522223620062954783, 4.62029792875819667985293043945, 6.03815345921504085971535402317, 6.99716998511362283849988254925, 8.009143662998636406389413906742, 9.169412126763078839012244728100, 9.812380172183782284393389591647, 10.88504989104748219586023649587, 12.00612755070628790159896884122

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