Properties

Label 2-273-91.20-c1-0-12
Degree $2$
Conductor $273$
Sign $0.999 + 0.0182i$
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  + (1.96 − 0.525i)2-s + (0.866 + 0.5i)3-s + (1.84 − 1.06i)4-s + (−1.56 + 1.56i)5-s + (1.96 + 0.525i)6-s + (2.60 − 0.462i)7-s + (0.187 − 0.187i)8-s + (0.499 + 0.866i)9-s + (−2.25 + 3.90i)10-s + (−1.16 − 4.33i)11-s + 2.13·12-s + (3.51 + 0.786i)13-s + (4.87 − 2.27i)14-s + (−2.14 + 0.573i)15-s + (−1.86 + 3.22i)16-s + (−1.31 − 2.27i)17-s + ⋯
L(s)  = 1  + (1.38 − 0.371i)2-s + (0.499 + 0.288i)3-s + (0.922 − 0.532i)4-s + (−0.700 + 0.700i)5-s + (0.801 + 0.214i)6-s + (0.984 − 0.174i)7-s + (0.0661 − 0.0661i)8-s + (0.166 + 0.288i)9-s + (−0.712 + 1.23i)10-s + (−0.350 − 1.30i)11-s + 0.614·12-s + (0.975 + 0.218i)13-s + (1.30 − 0.609i)14-s + (−0.552 + 0.148i)15-s + (−0.465 + 0.805i)16-s + (−0.318 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0182i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 + 0.0182i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69856 - 0.0245613i\)
\(L(\frac12)\) \(\approx\) \(2.69856 - 0.0245613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 + 0.462i)T \)
13 \( 1 + (-3.51 - 0.786i)T \)
good2 \( 1 + (-1.96 + 0.525i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.56 - 1.56i)T - 5iT^{2} \)
11 \( 1 + (1.16 + 4.33i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.31 + 2.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.01 + 1.61i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.58 + 2.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.06 - 3.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.44 + 2.44i)T - 31iT^{2} \)
37 \( 1 + (-0.0290 - 0.108i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.03 + 7.58i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.47 - 3.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.82 - 5.82i)T + 47iT^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 + (1.27 - 4.77i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.33 + 1.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 2.72i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.56 + 13.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11.8 - 11.8i)T + 73iT^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + (5.15 - 5.15i)T - 83iT^{2} \)
89 \( 1 + (7.93 - 2.12i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.93 - 0.518i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.80646585451561328739061228382, −11.03447419781760411134543476465, −10.72494622716124405984687340333, −8.751118590922303376840543963037, −8.143993032175343958706853859049, −6.72774859980331412266746123956, −5.55237141343924786492834515170, −4.30243995744353610830285541886, −3.60788865905239994869674158651, −2.40762633566964620028161636373, 2.05032080286107293197755202366, 3.93167706364877074818568685190, 4.47284697737346018951651105227, 5.63866310918900969108296038162, 6.83773876804828190454180387093, 8.025411834224175715318523032164, 8.564671199281382903021425860593, 10.08672267566886222687239748235, 11.47179627087538665665216608668, 12.26188142082340300551410628598

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