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Properties

Label 2-273-273.44-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.374 + 0.927i$
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 + (−0.753 + 1.55i)3-s + (−1.67 + 0.967i)4-s + (−2.71 + 0.728i)5-s + (0.0827 − 0.433i)6-s + (1.41 − 2.23i)7-s + (0.709 − 0.709i)8-s + (−1.86 − 2.35i)9-s + (0.620 − 0.358i)10-s + (3.10 + 0.831i)11-s + (−0.245 − 3.34i)12-s + (−3.47 + 0.968i)13-s + (−0.199 + 0.643i)14-s + (0.913 − 4.78i)15-s + (1.80 − 3.13i)16-s + (−3.96 − 6.85i)17-s + ⋯
L(s)  = 1  + (−0.174 + 0.0466i)2-s + (−0.435 + 0.900i)3-s + (−0.837 + 0.483i)4-s + (−1.21 + 0.325i)5-s + (0.0337 − 0.177i)6-s + (0.533 − 0.845i)7-s + (0.250 − 0.250i)8-s + (−0.621 − 0.783i)9-s + (0.196 − 0.113i)10-s + (0.935 + 0.250i)11-s + (−0.0708 − 0.964i)12-s + (−0.963 + 0.268i)13-s + (−0.0534 + 0.172i)14-s + (0.235 − 1.23i)15-s + (0.451 − 0.782i)16-s + (−0.960 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.374 + 0.927i$
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: \((2,\ 273,\ (\ :1/2),\ -0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0494014 - 0.0732252i\)
\(L(\frac12)\) \(\approx\) \(0.0494014 - 0.0732252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.753 - 1.55i)T \)
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

−11.67020122103647717119530839782, −10.74201061157658445058254035261, −9.582733674560965259393489312327, −8.941966223817681845767142375019, −7.61366590567285773703436863181, −7.00668419793757101363379900862, −4.96319903345641059733424425568, −4.30407149977493560039228428832, −3.48237946442833870747521969899, −0.07767388492667420286391326198, 1.77446346733959982595391832315, 4.02213531684840222845121497042, 5.10998847267230375288636516178, 6.15350781063778278803322019616, 7.48475925008510597766864560001, 8.493349219618373975006339982184, 8.892353624780685310071171346693, 10.55006184846932066588562437410, 11.37032712355532828897076499387, 12.35606612032108632361540940397

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