Properties

Label 2-273-91.24-c1-0-15
Degree $2$
Conductor $273$
Sign $0.928 + 0.370i$
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  + (2.45 − 0.657i)2-s + i·3-s + (3.84 − 2.22i)4-s + (−2.20 − 0.590i)5-s + (0.657 + 2.45i)6-s + (2.58 − 0.540i)7-s + (4.38 − 4.38i)8-s − 9-s − 5.79·10-s + (−1.22 + 1.22i)11-s + (2.22 + 3.84i)12-s + (1.05 + 3.44i)13-s + (5.99 − 3.02i)14-s + (0.590 − 2.20i)15-s + (3.43 − 5.94i)16-s + (−3.75 − 6.51i)17-s + ⋯
L(s)  = 1  + (1.73 − 0.464i)2-s + 0.577i·3-s + (1.92 − 1.11i)4-s + (−0.986 − 0.264i)5-s + (0.268 + 1.00i)6-s + (0.978 − 0.204i)7-s + (1.55 − 1.55i)8-s − 0.333·9-s − 1.83·10-s + (−0.370 + 0.370i)11-s + (0.641 + 1.11i)12-s + (0.292 + 0.956i)13-s + (1.60 − 0.809i)14-s + (0.152 − 0.569i)15-s + (0.858 − 1.48i)16-s + (−0.911 − 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.928 + 0.370i$
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: \((2,\ 273,\ (\ :1/2),\ 0.928 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86123 - 0.549480i\)
\(L(\frac12)\) \(\approx\) \(2.86123 - 0.549480i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.540i)T \)
13 \( 1 + (-1.05 - 3.44i)T \)
good2 \( 1 + (-2.45 + 0.657i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.20 + 0.590i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.22 - 1.22i)T - 11iT^{2} \)
17 \( 1 + (3.75 + 6.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.24 - 4.24i)T - 19iT^{2} \)
23 \( 1 + (-0.0187 - 0.0108i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.432 - 0.749i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.55 - 5.81i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.43 + 9.07i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.45 - 0.656i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.71 - 0.987i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.50 + 5.62i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.87 - 11.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.50 + 5.60i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 3.60iT - 61T^{2} \)
67 \( 1 + (0.794 + 0.794i)T + 67iT^{2} \)
71 \( 1 + (-12.5 + 3.36i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-9.58 + 2.56i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.16 - 3.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.64 + 8.64i)T - 83iT^{2} \)
89 \( 1 + (-2.34 + 0.627i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.99 - 7.43i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.91973546024497083178090919041, −11.19022117289561346741892417272, −10.60276529996689841442002501336, −9.034281985403626544333304512770, −7.74826838062945372122057071591, −6.60413578483411169753758256973, −5.14359818236141435664162519190, −4.48700233151285611968123219754, −3.78251412447087255770767201931, −2.18425199678450302841695494071, 2.40694813179278613117007931205, 3.74696797857926903120853730554, 4.72055465019578399600392445183, 5.87571167944922793034751400936, 6.75567597403594888543690710795, 7.943884731573461961948163350188, 8.323081451668458545550990374289, 10.89500098358050522394466746061, 11.23537165954499231611255904523, 12.22873735153953446610895774166

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