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Properties

Label 2-273-13.10-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.969 - 0.246i$
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  + (−2.20 + 1.27i)2-s + (0.5 + 0.866i)3-s + (2.23 − 3.86i)4-s + 4.07i·5-s + (−2.20 − 1.27i)6-s + (0.866 + 0.5i)7-s + 6.26i·8-s + (−0.499 + 0.866i)9-s + (−5.17 − 8.96i)10-s + (3.35 − 1.93i)11-s + 4.46·12-s + (1.02 + 3.45i)13-s − 2.54·14-s + (−3.52 + 2.03i)15-s + (−3.49 − 6.05i)16-s + (−1.23 + 2.14i)17-s + ⋯
L(s)  = 1  + (−1.55 + 0.898i)2-s + (0.288 + 0.499i)3-s + (1.11 − 1.93i)4-s + 1.82i·5-s + (−0.898 − 0.518i)6-s + (0.327 + 0.188i)7-s + 2.21i·8-s + (−0.166 + 0.288i)9-s + (−1.63 − 2.83i)10-s + (1.01 − 0.584i)11-s + 1.28·12-s + (0.283 + 0.959i)13-s − 0.679·14-s + (−0.910 + 0.525i)15-s + (−0.874 − 1.51i)16-s + (−0.299 + 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.969 - 0.246i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0842907 + 0.672772i\)
\(L(\frac12)\) \(\approx\) \(0.0842907 + 0.672772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-1.02 - 3.45i)T \)
good2 \( 1 + (2.20 - 1.27i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 4.07iT - 5T^{2} \)
11 \( 1 + (-3.35 + 1.93i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.23 - 2.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.32 - 1.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.42 + 4.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.88 + 4.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.35iT - 31T^{2} \)
37 \( 1 + (1.95 - 1.12i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.19 - 2.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.84 + 3.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (7.27 + 4.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.51 + 6.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.44 + 2.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.49 - 4.90i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.54iT - 73T^{2} \)
79 \( 1 - 0.0251T + 79T^{2} \)
83 \( 1 + 0.202iT - 83T^{2} \)
89 \( 1 + (-15.8 + 9.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.20 + 5.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.57232961828352234661852100240, −11.08280941108370665783705763049, −10.18351881219909165351876966719, −9.449925410547301672891292560063, −8.498377396524481258894404119980, −7.57632536480805782466830091379, −6.55855468054283324267664738822, −6.03703627879530081482819264777, −3.82885695997538386394059774834, −2.12430017928522832265562381455, 0.888676111604621731149161307124, 1.81149533727705152539289343249, 3.66944290724238377776534046624, 5.24306173581867180645520026376, 7.13629325583740142989024128483, 7.989613264403747331231265809520, 8.865650835629061584415243176630, 9.253506332157703488644865553925, 10.29937678098311451313039475557, 11.62817761852977539869481551368

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