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Properties

Label 2-273-91.81-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.0629 + 0.998i$
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  + (0.707 − 1.22i)2-s + 3-s + (−0.000924 − 0.00160i)4-s + (−1.42 − 2.47i)5-s + (0.707 − 1.22i)6-s + (−1.85 − 1.88i)7-s + 2.82·8-s + 9-s − 4.04·10-s + 2.30·11-s + (−0.000924 − 0.00160i)12-s + (−3.12 − 1.79i)13-s + (−3.62 + 0.940i)14-s + (−1.42 − 2.47i)15-s + (2.00 − 3.46i)16-s + (3.72 + 6.45i)17-s + ⋯
L(s)  = 1  + (0.500 − 0.866i)2-s + 0.577·3-s + (−0.000462 − 0.000800i)4-s + (−0.639 − 1.10i)5-s + (0.288 − 0.500i)6-s + (−0.701 − 0.712i)7-s + 0.999·8-s + 0.333·9-s − 1.27·10-s + 0.695·11-s + (−0.000266 − 0.000462i)12-s + (−0.866 − 0.498i)13-s + (−0.968 + 0.251i)14-s + (−0.369 − 0.639i)15-s + (0.500 − 0.866i)16-s + (0.903 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0629 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0629 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0629 + 0.998i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0629 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26104 - 1.34309i\)
\(L(\frac12)\) \(\approx\) \(1.26104 - 1.34309i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (1.85 + 1.88i)T \)
13 \( 1 + (3.12 + 1.79i)T \)
good2 \( 1 + (-0.707 + 1.22i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.42 + 2.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.30T + 11T^{2} \)
17 \( 1 + (-3.72 - 6.45i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.565T + 19T^{2} \)
23 \( 1 + (-0.398 + 0.690i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.00 - 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.80 - 6.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.15 - 1.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.68 + 9.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.134 - 0.233i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.35 + 4.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.91 - 5.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.93T + 61T^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
71 \( 1 + (3.01 - 5.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.31 + 9.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.01 + 3.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 + (-0.709 + 1.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.23 - 9.05i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.20452425840417970256630604085, −10.69104380730931327955756540180, −10.00192128889238320325407910640, −8.755423517737290183600119173177, −7.889486472172463773781730229544, −6.90343149445589215462185188373, −5.04536893337056189989637878456, −3.95667165990781742031183600703, −3.27259126997756811818348426862, −1.38912181451704014697826601803, 2.54203675773628175284012711181, 3.73766589477233330310191133001, 5.13692041924944482404005598680, 6.43062232975954269740392569789, 7.10379436206233221816033932626, 7.85482522478902386791545005893, 9.371370111899573976969212627251, 10.03281643325298069773951551325, 11.41958736759165375073588109743, 12.07685477569764887961990087353

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