Properties

Label 2-273-91.24-c1-0-6
Degree $2$
Conductor $273$
Sign $0.673 + 0.738i$
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  + (−1.53 + 0.409i)2-s + i·3-s + (0.440 − 0.254i)4-s + (−3.63 − 0.973i)5-s + (−0.409 − 1.53i)6-s + (−1.31 + 2.29i)7-s + (1.66 − 1.66i)8-s − 9-s + 5.95·10-s + (2.38 − 2.38i)11-s + (0.254 + 0.440i)12-s + (3.03 − 1.94i)13-s + (1.06 − 4.05i)14-s + (0.973 − 3.63i)15-s + (−2.37 + 4.12i)16-s + (−1.66 − 2.89i)17-s + ⋯
L(s)  = 1  + (−1.08 + 0.289i)2-s + 0.577i·3-s + (0.220 − 0.127i)4-s + (−1.62 − 0.435i)5-s + (−0.167 − 0.624i)6-s + (−0.495 + 0.868i)7-s + (0.590 − 0.590i)8-s − 0.333·9-s + 1.88·10-s + (0.718 − 0.718i)11-s + (0.0734 + 0.127i)12-s + (0.842 − 0.538i)13-s + (0.284 − 1.08i)14-s + (0.251 − 0.938i)15-s + (−0.594 + 1.03i)16-s + (−0.404 − 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.673 + 0.738i$
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: \((2,\ 273,\ (\ :1/2),\ 0.673 + 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308924 - 0.136381i\)
\(L(\frac12)\) \(\approx\) \(0.308924 - 0.136381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 - 2.29i)T \)
13 \( 1 + (-3.03 + 1.94i)T \)
good2 \( 1 + (1.53 - 0.409i)T + (1.73 - i)T^{2} \)
5 \( 1 + (3.63 + 0.973i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.38 + 2.38i)T - 11iT^{2} \)
17 \( 1 + (1.66 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.537 + 0.537i)T - 19iT^{2} \)
23 \( 1 + (-3.08 - 1.78i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.42 + 2.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.38 + 8.90i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.93 + 10.9i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.84 + 1.83i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.9 - 6.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.356 + 1.32i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.59 - 6.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.816 + 3.04i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 1.04iT - 61T^{2} \)
67 \( 1 + (2.09 + 2.09i)T + 67iT^{2} \)
71 \( 1 + (7.00 - 1.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.08 + 0.559i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.54 + 9.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.51 + 1.51i)T - 83iT^{2} \)
89 \( 1 + (3.05 - 0.819i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.23 - 12.0i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.45817947631143374472700208271, −10.92003401563759587376584769397, −9.374689537894174051866955498044, −8.966349676564580254713339189044, −8.167283672444203108594807022115, −7.25964971461396850407892470713, −5.83146710645449170939540802204, −4.29149881835076356734213134237, −3.39802601287147837378254843782, −0.42922094199379750881522164146, 1.31304660718358167479177622144, 3.45334567470657282524513875339, 4.51048838128762082693988418452, 6.75024688056930113658654408566, 7.19699794608416569745469604749, 8.279946827755491045146963891181, 8.955377817781037839757066708547, 10.29791032229151092454244337958, 10.98270091913392027410986371661, 11.75296762677755645527079551954

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