Properties

Label 2-273-91.19-c1-0-7
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: $\chi_{273} (19, \cdot )$
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

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

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