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Properties

Label 2-273-91.4-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.752 - 0.658i$
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  − 2.50i·2-s + (−0.5 − 0.866i)3-s − 4.27·4-s + (2.61 − 1.50i)5-s + (−2.17 + 1.25i)6-s + (−2.46 − 0.967i)7-s + 5.71i·8-s + (−0.499 + 0.866i)9-s + (−3.77 − 6.54i)10-s + (−1.34 + 0.775i)11-s + (2.13 + 3.70i)12-s + (−0.822 − 3.51i)13-s + (−2.42 + 6.17i)14-s + (−2.61 − 1.50i)15-s + 5.75·16-s − 1.51·17-s + ⋯
L(s)  = 1  − 1.77i·2-s + (−0.288 − 0.499i)3-s − 2.13·4-s + (1.16 − 0.674i)5-s + (−0.885 + 0.511i)6-s + (−0.930 − 0.365i)7-s + 2.01i·8-s + (−0.166 + 0.288i)9-s + (−1.19 − 2.07i)10-s + (−0.405 + 0.233i)11-s + (0.617 + 1.06i)12-s + (−0.228 − 0.973i)13-s + (−0.647 + 1.64i)14-s + (−0.674 − 0.389i)15-s + 1.43·16-s − 0.367·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350816 + 0.934475i\)
\(L(\frac12)\) \(\approx\) \(0.350816 + 0.934475i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.967i)T \)
13 \( 1 + (0.822 + 3.51i)T \)
good2 \( 1 + 2.50iT - 2T^{2} \)
5 \( 1 + (-2.61 + 1.50i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.34 - 0.775i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.51T + 17T^{2} \)
19 \( 1 + (-7.11 - 4.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.64T + 23T^{2} \)
29 \( 1 + (-2.75 + 4.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.47 + 2.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.20iT - 37T^{2} \)
41 \( 1 + (-2.85 - 1.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.97 + 3.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.00 - 4.62i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.90 + 6.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.0iT - 59T^{2} \)
61 \( 1 + (-5.23 + 9.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.52 + 1.45i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.11 - 0.642i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.27 + 1.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.28 + 5.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.49iT - 83T^{2} \)
89 \( 1 - 9.76iT - 89T^{2} \)
97 \( 1 + (-10.8 + 6.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.37565573882444402786668441368, −10.25887979298822282689080633231, −9.819668297898240480769286521156, −8.997762142927361446757384500892, −7.56072946615089032738101035001, −5.90119978802052936127269983752, −5.02645686009900961765896723464, −3.42258882623789690383900363650, −2.21142471867404603474950576375, −0.820742982557997120856144643589, 3.05936861474093561404332687224, 4.88542526965277326749814010802, 5.64584847945194774401828816156, 6.62869682663611812415409816445, 7.07347625404973294241150119141, 8.761348226052460960209166197352, 9.430898657073628037693510089098, 10.10487555813725256582607877382, 11.45005289362268256176495572549, 12.98469094099328640977951172222

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