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Properties

Label 2-273-91.80-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.967 + 0.253i$
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  + (−0.339 − 1.26i)2-s − i·3-s + (0.246 − 0.142i)4-s + (0.109 − 0.409i)5-s + (−1.26 + 0.339i)6-s + (−2.64 − 0.0688i)7-s + (−2.11 − 2.11i)8-s − 9-s − 0.555·10-s + (−0.991 − 0.991i)11-s + (−0.142 − 0.246i)12-s + (3.54 − 0.648i)13-s + (0.809 + 3.36i)14-s + (−0.409 − 0.109i)15-s + (−1.67 + 2.90i)16-s + (−3.47 − 6.01i)17-s + ⋯
L(s)  = 1  + (−0.239 − 0.894i)2-s − 0.577i·3-s + (0.123 − 0.0710i)4-s + (0.0491 − 0.183i)5-s + (−0.516 + 0.138i)6-s + (−0.999 − 0.0260i)7-s + (−0.748 − 0.748i)8-s − 0.333·9-s − 0.175·10-s + (−0.298 − 0.298i)11-s + (−0.0410 − 0.0710i)12-s + (0.983 − 0.179i)13-s + (0.216 + 0.900i)14-s + (−0.105 − 0.0283i)15-s + (−0.418 + 0.725i)16-s + (−0.842 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.967 + 0.253i$
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: \((2,\ 273,\ (\ :1/2),\ -0.967 + 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127049 - 0.987690i\)
\(L(\frac12)\) \(\approx\) \(0.127049 - 0.987690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0688i)T \)
13 \( 1 + (-3.54 + 0.648i)T \)
good2 \( 1 + (0.339 + 1.26i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.109 + 0.409i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.991 + 0.991i)T + 11iT^{2} \)
17 \( 1 + (3.47 + 6.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.391 - 0.391i)T + 19iT^{2} \)
23 \( 1 + (-6.79 - 3.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.01 + 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.09 - 2.17i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.79 + 1.55i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.434 + 1.62i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.49 - 3.74i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.79 - 2.62i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.77 + 6.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.60 + 1.76i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.75iT - 61T^{2} \)
67 \( 1 + (-10.1 + 10.1i)T - 67iT^{2} \)
71 \( 1 + (-1.00 - 3.74i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.96 + 11.0i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.32 - 7.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.26 - 2.26i)T + 83iT^{2} \)
89 \( 1 + (-2.99 - 11.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.5 - 4.17i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.17933991500793928220798493327, −10.94747740581572402891800896669, −9.404297706104456323385100468750, −9.092860632456871520750387338724, −7.42644990553408352871641135825, −6.55296325585237537006897496197, −5.50173244055385144280329192550, −3.53775202338132090183501530794, −2.53947225867514674867691211179, −0.820472095228341847157182560469, 2.71138599501762503623056061669, 4.01032151702853240486584993091, 5.61426164388197792808884434537, 6.45434761799191995607924730700, 7.27470197665246448015741626421, 8.692494254350650692148129897361, 9.080610300852050561002774038076, 10.58265002291160704295118763450, 11.03648568171146872116908244056, 12.51012693185558252922512631150

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