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Properties

Label 2-273-91.24-c1-0-13
Degree $2$
Conductor $273$
Sign $0.996 + 0.0887i$
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  + (2.02 − 0.542i)2-s + i·3-s + (2.07 − 1.19i)4-s + (1.89 + 0.507i)5-s + (0.542 + 2.02i)6-s + (−1.04 − 2.43i)7-s + (0.588 − 0.588i)8-s − 9-s + 4.10·10-s + (1.77 − 1.77i)11-s + (1.19 + 2.07i)12-s + (−3.33 + 1.36i)13-s + (−3.43 − 4.35i)14-s + (−0.507 + 1.89i)15-s + (−1.52 + 2.64i)16-s + (3.66 + 6.34i)17-s + ⋯
L(s)  = 1  + (1.43 − 0.383i)2-s + 0.577i·3-s + (1.03 − 0.599i)4-s + (0.846 + 0.226i)5-s + (0.221 + 0.826i)6-s + (−0.394 − 0.919i)7-s + (0.207 − 0.207i)8-s − 0.333·9-s + 1.29·10-s + (0.534 − 0.534i)11-s + (0.345 + 0.599i)12-s + (−0.925 + 0.378i)13-s + (−0.916 − 1.16i)14-s + (−0.130 + 0.488i)15-s + (−0.381 + 0.660i)16-s + (0.887 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.996 + 0.0887i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.996 + 0.0887i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67102 - 0.118705i\)
\(L(\frac12)\) \(\approx\) \(2.67102 - 0.118705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (1.04 + 2.43i)T \)
13 \( 1 + (3.33 - 1.36i)T \)
good2 \( 1 + (-2.02 + 0.542i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.89 - 0.507i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.77 + 1.77i)T - 11iT^{2} \)
17 \( 1 + (-3.66 - 6.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.681 + 0.681i)T - 19iT^{2} \)
23 \( 1 + (5.74 + 3.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.21 + 9.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.782 + 2.92i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.356 + 1.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.710 + 0.190i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.1 - 5.84i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.971 - 3.62i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.15 - 2.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.22 + 4.58i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 9.45iT - 61T^{2} \)
67 \( 1 + (-4.43 - 4.43i)T + 67iT^{2} \)
71 \( 1 + (-8.32 + 2.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.50 + 2.01i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.31 + 4.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 + (3.95 - 1.06i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.63 - 9.85i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.05343394262286727268661258202, −11.05023622104186973804158671216, −10.17575400500970757422871502613, −9.461729283585920785298913624360, −7.88357940624764497408084003364, −6.27194484305362689191769478135, −5.82559323411886185123594969396, −4.33747716016575458928683136720, −3.70244514693758235359881427323, −2.25934348846465236968097629069, 2.18576664647857346913799941566, 3.42152124048374019020346118999, 5.23432668550801675228764857893, 5.52416110335386467609004695450, 6.71232121372943668975516491298, 7.55102096562614096280959543959, 9.247916228288990017059952594622, 9.774822591570211886351502791307, 11.61005762597786121847002483699, 12.37811629174507555395437009961

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