Properties

Label 2-273-273.185-c1-0-25
Degree $2$
Conductor $273$
Sign $-0.139 + 0.990i$
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  + (1.21 + 0.702i)2-s + (−1.65 + 0.509i)3-s + (−0.0124 − 0.0215i)4-s + (−1.48 − 2.56i)5-s + (−2.37 − 0.543i)6-s + (−2.30 + 1.30i)7-s − 2.84i·8-s + (2.48 − 1.68i)9-s − 4.17i·10-s − 2.71i·11-s + (0.0315 + 0.0293i)12-s + (−3.51 + 0.787i)13-s + (−3.71 − 0.0247i)14-s + (3.76 + 3.49i)15-s + (1.97 − 3.42i)16-s + (−0.233 − 0.403i)17-s + ⋯
L(s)  = 1  + (0.860 + 0.496i)2-s + (−0.955 + 0.294i)3-s + (−0.00622 − 0.0107i)4-s + (−0.663 − 1.14i)5-s + (−0.968 − 0.221i)6-s + (−0.869 + 0.494i)7-s − 1.00i·8-s + (0.826 − 0.562i)9-s − 1.31i·10-s − 0.820i·11-s + (0.00911 + 0.00847i)12-s + (−0.975 + 0.218i)13-s + (−0.993 − 0.00661i)14-s + (0.972 + 0.903i)15-s + (0.493 − 0.855i)16-s + (−0.0565 − 0.0979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486483 - 0.559983i\)
\(L(\frac12)\) \(\approx\) \(0.486483 - 0.559983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 - 0.509i)T \)
7 \( 1 + (2.30 - 1.30i)T \)
13 \( 1 + (3.51 - 0.787i)T \)
good2 \( 1 + (-1.21 - 0.702i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.48 + 2.56i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
17 \( 1 + (0.233 + 0.403i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.58iT - 19T^{2} \)
23 \( 1 + (-2.09 - 1.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.22 - 4.16i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.80 - 3.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.67 + 9.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.19 + 7.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 3.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.26 - 7.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.995 - 0.574i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.19 + 8.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.12iT - 61T^{2} \)
67 \( 1 - 4.45T + 67T^{2} \)
71 \( 1 + (-7.31 - 4.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.75 + 3.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.45 + 2.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + (-3.44 + 5.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.95 - 5.17i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.99319007280601425083166763921, −10.90429152631222533159386131004, −9.589730940124386004067086477079, −9.013980055368332758626239250950, −7.36395028742124484706164628115, −6.32276618811861446047895160085, −5.36705560741592770605222961969, −4.71938690205660963322635174017, −3.58494733275252906778854417411, −0.47733504545606326179192302756, 2.56260971846784491078149251160, 3.80524240464492978812812846010, 4.77403963724817679677649308667, 6.14826516221523588896584764233, 7.13114712427303812401643860071, 7.83317154293109388891283655452, 9.863113803002139055067327792530, 10.50013379837840016051004195957, 11.58265598887944987442849508646, 12.01721691754822552994975979916

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