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Properties

Label 2-273-91.83-c1-0-5
Degree $2$
Conductor $273$
Sign $-0.776 - 0.630i$
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  + (−1.22 + 1.22i)2-s + i·3-s − 0.999i·4-s + (2 + 2i)5-s + (−1.22 − 1.22i)6-s + (2.44 + i)7-s + (−1.22 − 1.22i)8-s − 9-s − 4.89·10-s + (4.44 + 4.44i)11-s + 0.999·12-s + (2 − 3i)13-s + (−4.22 + 1.77i)14-s + (−2 + 2i)15-s + 5·16-s − 2·17-s + ⋯
L(s)  = 1  + (−0.866 + 0.866i)2-s + 0.577i·3-s − 0.499i·4-s + (0.894 + 0.894i)5-s + (−0.499 − 0.499i)6-s + (0.925 + 0.377i)7-s + (−0.433 − 0.433i)8-s − 0.333·9-s − 1.54·10-s + (1.34 + 1.34i)11-s + 0.288·12-s + (0.554 − 0.832i)13-s + (−1.12 + 0.474i)14-s + (−0.516 + 0.516i)15-s + 1.25·16-s − 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.776 - 0.630i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.776 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342096 + 0.964641i\)
\(L(\frac12)\) \(\approx\) \(0.342096 + 0.964641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 - i)T \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (1.22 - 1.22i)T - 2iT^{2} \)
5 \( 1 + (-2 - 2i)T + 5iT^{2} \)
11 \( 1 + (-4.44 - 4.44i)T + 11iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (5.44 + 5.44i)T + 19iT^{2} \)
23 \( 1 - 0.898iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 + (0.550 + 0.550i)T + 31iT^{2} \)
37 \( 1 + (1.89 + 1.89i)T + 37iT^{2} \)
41 \( 1 + (4 + 4i)T + 41iT^{2} \)
43 \( 1 + 2.89iT - 43T^{2} \)
47 \( 1 + (-6.44 + 6.44i)T - 47iT^{2} \)
53 \( 1 - 7.79T + 53T^{2} \)
59 \( 1 + (-0.449 + 0.449i)T - 59iT^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + (-8.34 + 8.34i)T - 67iT^{2} \)
71 \( 1 + (2.44 - 2.44i)T - 71iT^{2} \)
73 \( 1 + (1.89 - 1.89i)T - 73iT^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 + (4.44 + 4.44i)T + 83iT^{2} \)
89 \( 1 + (-2 + 2i)T - 89iT^{2} \)
97 \( 1 + (1.89 + 1.89i)T + 97iT^{2} \)
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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

−12.10824784513442719717283428760, −10.97239899934893016818879218654, −10.19112620707226680608573391229, −9.188225277833801210454641675831, −8.671268309925956163172455826375, −7.28689479760363063113275329645, −6.56945405687456100145596481715, −5.50422801341887080606689785023, −3.99929676968126252418787321044, −2.17541129203978597405439983548, 1.22317055223549272782744152085, 1.88392866495815513188344540944, 3.95692633474812991662861989877, 5.60823923314576919235415092196, 6.45526391812804286070420999543, 8.240001933471805795657655140962, 8.752204616803414073277977067793, 9.463905075170808674064081974065, 10.76466002958575485270287729511, 11.36416349570244497801601036296

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