Properties

Label 2-273-21.5-c1-0-16
Degree $2$
Conductor $273$
Sign $-0.563 + 0.826i$
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  + (−1.59 − 0.918i)2-s + (−0.540 + 1.64i)3-s + (0.686 + 1.18i)4-s + (1.90 − 3.30i)5-s + (2.36 − 2.12i)6-s + (−2.46 + 0.954i)7-s + 1.15i·8-s + (−2.41 − 1.77i)9-s + (−6.06 + 3.50i)10-s + (2.78 − 1.60i)11-s + (−2.32 + 0.487i)12-s + i·13-s + (4.80 + 0.746i)14-s + (4.40 + 4.92i)15-s + (2.43 − 4.21i)16-s + (−2.84 − 4.92i)17-s + ⋯
L(s)  = 1  + (−1.12 − 0.649i)2-s + (−0.311 + 0.950i)3-s + (0.343 + 0.594i)4-s + (0.852 − 1.47i)5-s + (0.967 − 0.866i)6-s + (−0.932 + 0.360i)7-s + 0.407i·8-s + (−0.805 − 0.592i)9-s + (−1.91 + 1.10i)10-s + (0.840 − 0.484i)11-s + (−0.671 + 0.140i)12-s + 0.277i·13-s + (1.28 + 0.199i)14-s + (1.13 + 1.27i)15-s + (0.607 − 1.05i)16-s + (−0.689 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.563 + 0.826i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227768 - 0.430826i\)
\(L(\frac12)\) \(\approx\) \(0.227768 - 0.430826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.540 - 1.64i)T \)
7 \( 1 + (2.46 - 0.954i)T \)
13 \( 1 - iT \)
good2 \( 1 + (1.59 + 0.918i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.90 + 3.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.78 + 1.60i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.84 + 4.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.11 + 1.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.63 + 2.67i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.66iT - 29T^{2} \)
31 \( 1 + (-4.45 + 2.57i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.861 - 1.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 + (-3.80 + 6.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.76 - 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.53 - 11.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.115 + 0.0666i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.64 - 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.896iT - 71T^{2} \)
73 \( 1 + (0.619 - 0.357i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.40 - 9.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.58T + 83T^{2} \)
89 \( 1 + (-6.82 + 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.46iT - 97T^{2} \)
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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

−11.58872154342264107084777679281, −10.13398146338038780117559194845, −9.729706485072517814616380280579, −8.885788250056908998466567414771, −8.612254427218915316610822638258, −6.38274476888848991413484439002, −5.42570066079824321062744953549, −4.27439419232751208277884378297, −2.42868817765506935690991589012, −0.54772365764052676462117967574, 1.83363162256543356668734065118, 3.50340835034046346685326190139, 6.08619376711195019570617660497, 6.54256371832142802880896955842, 7.10827588872524832921313152681, 8.204467342183940851876997153488, 9.392459394235300124421825363485, 10.26111263464220517395147858406, 10.85207359054637682570861467292, 12.30955291909565117862417654724

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