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Properties

Label 2-273-91.31-c1-0-15
Degree $2$
Conductor $273$
Sign $0.984 - 0.173i$
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  + (1.71 + 0.458i)2-s + (0.866 + 0.5i)3-s + (0.987 + 0.570i)4-s + (0.586 − 2.18i)5-s + (1.25 + 1.25i)6-s + (2.59 − 0.537i)7-s + (−1.07 − 1.07i)8-s + (0.499 + 0.866i)9-s + (2.00 − 3.47i)10-s + (−3.49 + 0.937i)11-s + (0.570 + 0.987i)12-s + (−1.56 + 3.24i)13-s + (4.68 + 0.268i)14-s + (1.60 − 1.60i)15-s + (−2.49 − 4.31i)16-s + (−2.54 + 4.40i)17-s + ⋯
L(s)  = 1  + (1.21 + 0.324i)2-s + (0.499 + 0.288i)3-s + (0.493 + 0.285i)4-s + (0.262 − 0.979i)5-s + (0.511 + 0.511i)6-s + (0.979 − 0.202i)7-s + (−0.380 − 0.380i)8-s + (0.166 + 0.288i)9-s + (0.635 − 1.09i)10-s + (−1.05 + 0.282i)11-s + (0.164 + 0.285i)12-s + (−0.435 + 0.900i)13-s + (1.25 + 0.0718i)14-s + (0.413 − 0.413i)15-s + (−0.622 − 1.07i)16-s + (−0.616 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58500 + 0.226538i\)
\(L(\frac12)\) \(\approx\) \(2.58500 + 0.226538i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 0.537i)T \)
13 \( 1 + (1.56 - 3.24i)T \)
good2 \( 1 + (-1.71 - 0.458i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.586 + 2.18i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.49 - 0.937i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.54 - 4.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.781 - 2.91i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.54 + 1.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.91T + 29T^{2} \)
31 \( 1 + (2.64 - 0.708i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.43 + 5.36i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.23 + 5.23i)T + 41iT^{2} \)
43 \( 1 - 5.01iT - 43T^{2} \)
47 \( 1 + (5.79 + 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.73 + 9.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.328 + 1.22i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-8.11 + 4.68i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.04 + 7.63i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.84 + 1.84i)T - 71iT^{2} \)
73 \( 1 + (-2.48 - 9.26i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.0621 - 0.107i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.88 - 9.88i)T + 83iT^{2} \)
89 \( 1 + (17.5 + 4.71i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.0 - 10.0i)T + 97iT^{2} \)
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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

−12.40538821456340338970420261700, −11.14504163496759564072951343567, −10.00642521155463089840730266666, −8.918513243864849388951250161205, −8.102675339353607030653117408592, −6.82804846326057153478670293365, −5.37069624356918278628183598791, −4.78826582464888998434377144031, −3.89821638389978334314704189852, −2.06219731988456224870248832703, 2.48615361788570981868333915277, 3.01609450787382966708811241760, 4.69485378404855468630903516530, 5.49200965677929660751543012299, 6.84282779448854064548737645071, 7.88656463415871541702944647643, 8.892259519412399095404011615290, 10.34542494574368904425323947963, 11.16354039049487450728795935438, 11.94093393761948209997449156204

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