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Properties

Label 2-273-91.24-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.215 - 0.976i$
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  + (−1.46 + 0.393i)2-s − i·3-s + (0.271 − 0.156i)4-s + (−3.59 − 0.962i)5-s + (0.393 + 1.46i)6-s + (2.64 + 0.145i)7-s + (1.81 − 1.81i)8-s − 9-s + 5.65·10-s + (−1.41 + 1.41i)11-s + (−0.156 − 0.271i)12-s + (−1.45 + 3.29i)13-s + (−3.93 + 0.825i)14-s + (−0.962 + 3.59i)15-s + (−2.26 + 3.92i)16-s + (2.36 + 4.08i)17-s + ⋯
L(s)  = 1  + (−1.03 + 0.278i)2-s − 0.577i·3-s + (0.135 − 0.0784i)4-s + (−1.60 − 0.430i)5-s + (0.160 + 0.599i)6-s + (0.998 + 0.0551i)7-s + (0.641 − 0.641i)8-s − 0.333·9-s + 1.78·10-s + (−0.425 + 0.425i)11-s + (−0.0453 − 0.0784i)12-s + (−0.404 + 0.914i)13-s + (−1.05 + 0.220i)14-s + (−0.248 + 0.927i)15-s + (−0.566 + 0.980i)16-s + (0.572 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.215 - 0.976i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.215 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202421 + 0.251945i\)
\(L(\frac12)\) \(\approx\) \(0.202421 + 0.251945i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.145i)T \)
13 \( 1 + (1.45 - 3.29i)T \)
good2 \( 1 + (1.46 - 0.393i)T + (1.73 - i)T^{2} \)
5 \( 1 + (3.59 + 0.962i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.41 - 1.41i)T - 11iT^{2} \)
17 \( 1 + (-2.36 - 4.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.15 - 3.15i)T - 19iT^{2} \)
23 \( 1 + (1.80 + 1.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.10 - 8.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.591 + 2.20i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.166 - 0.622i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.81 + 2.36i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0966 - 0.0557i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.16 - 4.33i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.18 - 3.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.438 + 1.63i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 6.75iT - 61T^{2} \)
67 \( 1 + (2.10 + 2.10i)T + 67iT^{2} \)
71 \( 1 + (-14.4 + 3.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.1 - 2.72i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0273 + 0.0473i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.05 - 9.05i)T - 83iT^{2} \)
89 \( 1 + (6.91 - 1.85i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.20 + 11.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.25411308095204896409050097225, −11.19078254515597384025778519564, −10.27588999526425500524543129084, −8.829979732089738554215900191830, −8.193012479220699425807317025159, −7.71618355446490282535898692001, −6.77351307118733060984990605679, −4.85891562901534863401097653649, −3.91772883749006024367246047701, −1.53058333925469310328312395732, 0.37240472876280734758149127290, 2.82785961367279458901813460541, 4.32641287252846158129182248537, 5.21682671851314400067128988301, 7.26457728392492428465722927908, 8.106928169779890440278443248801, 8.493795190952837621380702123765, 9.917642873443906805315446408814, 10.64517712889761986288288404097, 11.41368424433287255282082746719

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