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Properties

Label 2-273-91.76-c1-0-13
Degree $2$
Conductor $273$
Sign $0.0992 + 0.995i$
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  + (0.385 − 1.43i)2-s + (0.866 − 0.5i)3-s + (−0.189 − 0.109i)4-s + (1.07 − 1.07i)5-s + (−0.385 − 1.43i)6-s + (−0.612 + 2.57i)7-s + (1.87 − 1.87i)8-s + (0.499 − 0.866i)9-s + (−1.12 − 1.95i)10-s + (1.68 + 0.451i)11-s − 0.218·12-s + (−3.51 − 0.818i)13-s + (3.46 + 1.87i)14-s + (0.392 − 1.46i)15-s + (−2.19 − 3.80i)16-s + (−1.43 + 2.48i)17-s + ⋯
L(s)  = 1  + (0.272 − 1.01i)2-s + (0.499 − 0.288i)3-s + (−0.0945 − 0.0545i)4-s + (0.479 − 0.479i)5-s + (−0.157 − 0.587i)6-s + (−0.231 + 0.972i)7-s + (0.663 − 0.663i)8-s + (0.166 − 0.288i)9-s + (−0.357 − 0.618i)10-s + (0.508 + 0.136i)11-s − 0.0630·12-s + (−0.973 − 0.227i)13-s + (0.926 + 0.500i)14-s + (0.101 − 0.378i)15-s + (−0.548 − 0.950i)16-s + (−0.348 + 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0992 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0992 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0992 + 0.995i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.0992 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41718 - 1.28284i\)
\(L(\frac12)\) \(\approx\) \(1.41718 - 1.28284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.612 - 2.57i)T \)
13 \( 1 + (3.51 + 0.818i)T \)
good2 \( 1 + (-0.385 + 1.43i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.07 + 1.07i)T - 5iT^{2} \)
11 \( 1 + (-1.68 - 0.451i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.43 - 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.389 + 1.45i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.21 - 1.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.65 + 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.32 - 1.32i)T - 31iT^{2} \)
37 \( 1 + (5.83 + 1.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.10 - 0.830i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.29 - 1.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.86 - 5.86i)T + 47iT^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 + (0.967 - 0.259i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.0305 - 0.0176i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 - 4.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.4 + 3.05i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 + 3.53T + 79T^{2} \)
83 \( 1 + (-10.8 + 10.8i)T - 83iT^{2} \)
89 \( 1 + (2.02 - 7.54i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.46 - 9.21i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.96838256458404563039653988682, −10.88267484873675296097123312217, −9.675417136274608100888351613063, −9.177734203310568341384295900689, −7.903115864810220136158541269309, −6.72562842501886059400511308686, −5.43247520529431795993948118997, −4.04017960310614269149470650609, −2.70051207807511346811701657986, −1.75923640512888759480806371774, 2.25315462391257015527620056409, 3.93140691205604465252972075249, 5.07332461692781355881876589065, 6.39819846954057235095894266742, 7.07477048463450594544948423022, 7.943997688371162475207520213583, 9.238981770604146413251791998742, 10.21564269715471497771236892344, 10.90968541136985362202627544996, 12.23675637254685555618268584153

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