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Properties

Label 2-273-91.80-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.965 + 0.259i$
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  + (0.705 + 2.63i)2-s − i·3-s + (−4.70 + 2.71i)4-s + (−0.914 + 3.41i)5-s + (2.63 − 0.705i)6-s + (2.62 − 0.362i)7-s + (−6.62 − 6.62i)8-s − 9-s − 9.64·10-s + (0.312 + 0.312i)11-s + (2.71 + 4.70i)12-s + (−3.60 + 0.0874i)13-s + (2.80 + 6.64i)14-s + (3.41 + 0.914i)15-s + (7.34 − 12.7i)16-s + (0.715 + 1.23i)17-s + ⋯
L(s)  = 1  + (0.499 + 1.86i)2-s − 0.577i·3-s + (−2.35 + 1.35i)4-s + (−0.409 + 1.52i)5-s + (1.07 − 0.288i)6-s + (0.990 − 0.137i)7-s + (−2.34 − 2.34i)8-s − 0.333·9-s − 3.04·10-s + (0.0942 + 0.0942i)11-s + (0.784 + 1.35i)12-s + (−0.999 + 0.0242i)13-s + (0.749 + 1.77i)14-s + (0.881 + 0.236i)15-s + (1.83 − 3.18i)16-s + (0.173 + 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.965 + 0.259i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.965 + 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164482 - 1.24503i\)
\(L(\frac12)\) \(\approx\) \(0.164482 - 1.24503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.362i)T \)
13 \( 1 + (3.60 - 0.0874i)T \)
good2 \( 1 + (-0.705 - 2.63i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.914 - 3.41i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.312 - 0.312i)T + 11iT^{2} \)
17 \( 1 + (-0.715 - 1.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.931 + 0.931i)T + 19iT^{2} \)
23 \( 1 + (-6.22 - 3.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.82 - 6.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.698 + 0.187i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.982 - 0.263i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.748 + 2.79i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.20 - 4.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.906 - 0.242i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.48 + 4.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.14 + 0.841i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 13.0iT - 61T^{2} \)
67 \( 1 + (-4.49 + 4.49i)T - 67iT^{2} \)
71 \( 1 + (-0.487 - 1.82i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.29 + 12.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.77 + 3.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.33 + 2.33i)T + 83iT^{2} \)
89 \( 1 + (2.70 + 10.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.51 + 0.673i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.69857504291248207393988055131, −11.70074832119352106642262761283, −10.55237164977011500170052485687, −9.059211949655963261178534311651, −7.965031248703986724136424307074, −7.24329919954545697643433949927, −6.84838345048811520837057127857, −5.61025164698421465405126459544, −4.50930222074361468743310429862, −3.09044683216902559785865209783, 0.897320066369147986918372616639, 2.49431517496025673433164248094, 4.13615462468224721690386811604, 4.75736504774665894631592147619, 5.40629049486089370846454104546, 8.126616110993377934562272224440, 8.913011257597270707499733217567, 9.645844230729352991415112894794, 10.68232163143246162637407994293, 11.55462635094217481189100046591

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