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Properties

Label 2-273-91.73-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.386 + 0.922i$
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  + (−0.517 + 1.93i)2-s + (0.866 + 0.5i)3-s + (−1.73 − 0.999i)4-s + (−3.58 − 0.960i)5-s + (−1.41 + 1.41i)6-s + (−2.59 + 0.534i)7-s + (−0.000695 + 0.000695i)8-s + (0.499 + 0.866i)9-s + (3.71 − 6.42i)10-s + (−1.11 − 4.17i)11-s + (−0.999 − 1.73i)12-s + (3.27 + 1.51i)13-s + (0.309 − 5.28i)14-s + (−2.62 − 2.62i)15-s + (−2.00 − 3.46i)16-s + (−3.45 + 5.98i)17-s + ⋯
L(s)  = 1  + (−0.366 + 1.36i)2-s + (0.499 + 0.288i)3-s + (−0.865 − 0.499i)4-s + (−1.60 − 0.429i)5-s + (−0.577 + 0.577i)6-s + (−0.979 + 0.201i)7-s + (−0.000246 + 0.000246i)8-s + (0.166 + 0.288i)9-s + (1.17 − 2.03i)10-s + (−0.336 − 1.25i)11-s + (−0.288 − 0.499i)12-s + (0.907 + 0.419i)13-s + (0.0827 − 1.41i)14-s + (−0.677 − 0.677i)15-s + (−0.500 − 0.866i)16-s + (−0.838 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144882 - 0.217834i\)
\(L(\frac12)\) \(\approx\) \(0.144882 - 0.217834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.59 - 0.534i)T \)
13 \( 1 + (-3.27 - 1.51i)T \)
good2 \( 1 + (0.517 - 1.93i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (3.58 + 0.960i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.11 + 4.17i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.45 - 5.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.46 + 0.927i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.10 - 0.635i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 + (-1.49 - 5.58i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.12 + 0.301i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.44 - 1.44i)T - 41iT^{2} \)
43 \( 1 + 2.89iT - 43T^{2} \)
47 \( 1 + (2.91 - 10.8i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.492 + 0.852i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-12.9 + 3.47i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (11.5 - 6.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.14 - 0.841i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.126 - 0.126i)T + 71iT^{2} \)
73 \( 1 + (6.02 - 1.61i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.82 + 3.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.62 + 2.62i)T - 83iT^{2} \)
89 \( 1 + (-2.79 + 10.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.18 - 3.18i)T - 97iT^{2} \)
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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

−12.74283312700008007268518747687, −11.52117285736272221110622509197, −10.63584747276648237573263965733, −8.935681482750303298509856101101, −8.631358851464897852384355458328, −7.909045491181708723998041918700, −6.74109044305776866031617729816, −5.86942952067069862515776506240, −4.28585826444528603943654376446, −3.31619636919869622683262233367, 0.20257379581784098193306639645, 2.41367740779193237765723672545, 3.47363596556355003084014880478, 4.26830918578015715823051766221, 6.63894036792949595874087488104, 7.45367678856464159923300594244, 8.553567453852163188825005194616, 9.548512526717012791219514969952, 10.43258989674339939353647749184, 11.30955968055163440276541745648

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