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Properties

Label 2-273-39.5-c1-0-9
Degree $2$
Conductor $273$
Sign $-0.605 - 0.795i$
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  + (1.65 + 1.65i)2-s + (0.818 + 1.52i)3-s + 3.46i·4-s + (−1.96 − 1.96i)5-s + (−1.17 + 3.87i)6-s + (0.707 + 0.707i)7-s + (−2.42 + 2.42i)8-s + (−1.66 + 2.49i)9-s − 6.48i·10-s + (−2.17 + 2.17i)11-s + (−5.28 + 2.83i)12-s + (2.86 − 2.18i)13-s + 2.33i·14-s + (1.39 − 4.60i)15-s − 1.07·16-s + 3.32·17-s + ⋯
L(s)  = 1  + (1.16 + 1.16i)2-s + (0.472 + 0.881i)3-s + 1.73i·4-s + (−0.877 − 0.877i)5-s + (−0.478 + 1.58i)6-s + (0.267 + 0.267i)7-s + (−0.855 + 0.855i)8-s + (−0.553 + 0.832i)9-s − 2.05i·10-s + (−0.655 + 0.655i)11-s + (−1.52 + 0.818i)12-s + (0.795 − 0.605i)13-s + 0.624i·14-s + (0.359 − 1.18i)15-s − 0.268·16-s + 0.807·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02637 + 2.07198i\)
\(L(\frac12)\) \(\approx\) \(1.02637 + 2.07198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.818 - 1.52i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-2.86 + 2.18i)T \)
good2 \( 1 + (-1.65 - 1.65i)T + 2iT^{2} \)
5 \( 1 + (1.96 + 1.96i)T + 5iT^{2} \)
11 \( 1 + (2.17 - 2.17i)T - 11iT^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 + (-2.91 + 2.91i)T - 19iT^{2} \)
23 \( 1 - 0.190T + 23T^{2} \)
29 \( 1 + 7.09iT - 29T^{2} \)
31 \( 1 + (-1.90 + 1.90i)T - 31iT^{2} \)
37 \( 1 + (6.97 + 6.97i)T + 37iT^{2} \)
41 \( 1 + (6.16 + 6.16i)T + 41iT^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 + (2.28 - 2.28i)T - 47iT^{2} \)
53 \( 1 - 5.76iT - 53T^{2} \)
59 \( 1 + (2.95 - 2.95i)T - 59iT^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (4.16 - 4.16i)T - 67iT^{2} \)
71 \( 1 + (1.60 + 1.60i)T + 71iT^{2} \)
73 \( 1 + (-0.722 - 0.722i)T + 73iT^{2} \)
79 \( 1 - 7.14T + 79T^{2} \)
83 \( 1 + (3.15 + 3.15i)T + 83iT^{2} \)
89 \( 1 + (-5.99 + 5.99i)T - 89iT^{2} \)
97 \( 1 + (7.27 - 7.27i)T - 97iT^{2} \)
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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

−12.50338662528422617874632477335, −11.61833137222244853779595767508, −10.31961693226683662605879170625, −9.024894489643429807406549299983, −8.013441220676467713671198912019, −7.60348244838549565218501638943, −5.83238109079960727010574539432, −4.97862219208454771740269442283, −4.27657638757524634102378106796, −3.17098392498293021533511343572, 1.53803947017575477522547357829, 3.23359407351902953787887923298, 3.50728367230084205699213823347, 5.21683147500454393544211206699, 6.49932693883514855694232841862, 7.59309384574878469167750568170, 8.539435738385662988683473418817, 10.20994806388404389308730890093, 11.01513144315781451433080753651, 11.77565085713233705562899695013

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