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Properties

Label 2-273-273.44-c1-0-8
Degree $2$
Conductor $273$
Sign $-0.0109 - 0.999i$
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  + (0.965 − 0.258i)2-s + (1.10 + 1.33i)3-s + (−0.866 + 0.5i)4-s + (−3.29 + 0.883i)5-s + (1.41 + i)6-s + (2.38 + 1.15i)7-s + (−2.12 + 2.12i)8-s + (−0.548 + 2.94i)9-s + (−2.95 + 1.70i)10-s + (−0.165 − 0.0444i)11-s + (−1.62 − 0.599i)12-s + (2 − 3i)13-s + (2.59 + 0.500i)14-s + (−4.82 − 3.41i)15-s + (−0.500 + 0.866i)16-s + (1.91 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.639 + 0.769i)3-s + (−0.433 + 0.250i)4-s + (−1.47 + 0.395i)5-s + (0.577 + 0.408i)6-s + (0.899 + 0.436i)7-s + (−0.749 + 0.749i)8-s + (−0.182 + 0.983i)9-s + (−0.935 + 0.539i)10-s + (−0.0499 − 0.0133i)11-s + (−0.469 − 0.173i)12-s + (0.554 − 0.832i)13-s + (0.694 + 0.133i)14-s + (−1.24 − 0.881i)15-s + (−0.125 + 0.216i)16-s + (0.464 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0109 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0109 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0109 - 0.999i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0109 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08638 + 1.09836i\)
\(L(\frac12)\) \(\approx\) \(1.08638 + 1.09836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.10 - 1.33i)T \)
7 \( 1 + (-2.38 - 1.15i)T \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (-0.965 + 0.258i)T + (1.73 - i)T^{2} \)
5 \( 1 + (3.29 - 0.883i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.165 + 0.0444i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.91 - 3.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.20 - 4.49i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.70 + 8.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + iT - 29T^{2} \)
31 \( 1 + (4.66 + 1.24i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.86 + 1.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4 - 4i)T + 41iT^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + (-2.75 - 10.2i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.30 + 1.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.23 + 1.93i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 + 0.902i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.53 - 7.53i)T + 71iT^{2} \)
73 \( 1 + (-1.85 + 6.92i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + (-1.16 - 4.33i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−12.22247504921918516392826141338, −11.18040174616141996132333586490, −10.59165448266666616662198064597, −9.028485979833851009898318790285, −8.141814200957845548563999744917, −7.82730220584081997677817457487, −5.69964795978486718992793557296, −4.55517910182417662272292542536, −3.80968548597464787013399957272, −2.86769731844695457502430209957, 1.01893818137153297736184366967, 3.38092728115542775821230650084, 4.27986889657400770003630959344, 5.32169669049950408943255325262, 7.00835718543543358273076696729, 7.59213455306160398302503523462, 8.701449038411044663030534481589, 9.361573896025000419110948444601, 11.25406147548404089194004182567, 11.73992410110790238579903489926

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