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Properties

Label 2-273-273.44-c1-0-21
Degree $2$
Conductor $273$
Sign $0.999 + 0.00670i$
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  + (0.246 − 0.0659i)2-s + (1.72 + 0.126i)3-s + (−1.67 + 0.967i)4-s + (2.71 − 0.728i)5-s + (0.433 − 0.0827i)6-s + (1.41 − 2.23i)7-s + (−0.709 + 0.709i)8-s + (2.96 + 0.438i)9-s + (0.620 − 0.358i)10-s + (−3.10 − 0.831i)11-s + (−3.01 + 1.45i)12-s + (−3.47 + 0.968i)13-s + (0.199 − 0.643i)14-s + (4.78 − 0.913i)15-s + (1.80 − 3.13i)16-s + (3.96 + 6.85i)17-s + ⋯
L(s)  = 1  + (0.174 − 0.0466i)2-s + (0.997 + 0.0731i)3-s + (−0.837 + 0.483i)4-s + (1.21 − 0.325i)5-s + (0.177 − 0.0337i)6-s + (0.533 − 0.845i)7-s + (−0.250 + 0.250i)8-s + (0.989 + 0.146i)9-s + (0.196 − 0.113i)10-s + (−0.935 − 0.250i)11-s + (−0.871 + 0.421i)12-s + (−0.963 + 0.268i)13-s + (0.0534 − 0.172i)14-s + (1.23 − 0.235i)15-s + (0.451 − 0.782i)16-s + (0.960 + 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.00670i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 + 0.00670i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87420 - 0.00628266i\)
\(L(\frac12)\) \(\approx\) \(1.87420 - 0.00628266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.126i)T \)
7 \( 1 + (-1.41 + 2.23i)T \)
13 \( 1 + (3.47 - 0.968i)T \)
good2 \( 1 + (-0.246 + 0.0659i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.71 + 0.728i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.10 + 0.831i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.96 - 6.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0189 + 0.0708i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.44 + 2.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0351iT - 29T^{2} \)
31 \( 1 + (8.23 + 2.20i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.86 - 2.37i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.39 + 1.39i)T + 41iT^{2} \)
43 \( 1 - 0.378iT - 43T^{2} \)
47 \( 1 + (0.516 + 1.92i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.88 - 3.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.27 - 2.21i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.83 - 8.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.85 + 1.30i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.74 - 1.74i)T + 71iT^{2} \)
73 \( 1 + (-2.08 + 7.76i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.79 + 6.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 - 3.46i)T + 83iT^{2} \)
89 \( 1 + (3.92 + 14.6i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.107 + 0.107i)T - 97iT^{2} \)
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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

−12.38548893339757850138472848944, −10.51765963514805221902345302721, −9.982322512048926300249814304207, −8.983379737811850305637557947688, −8.159425806328719392079062263381, −7.32465593467395853920734521007, −5.55196916772251076698945859443, −4.58367923067162462471798488123, −3.39520975028384968923188781583, −1.86330019707336049211102645687, 1.95596878655677683248455759078, 3.08453780406649696290071720891, 5.05910572038100561679066655040, 5.42980644895708916988100424097, 7.09480867033839064867125404847, 8.153257766378407903266594112647, 9.416698729625160102470949942110, 9.586875601777155021004072048623, 10.60945005198824552304685675722, 12.26010906028898045273491375980

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