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Properties

Label 2-273-91.31-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.693 - 0.720i$
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  + (−2.19 − 0.587i)2-s + (−0.866 − 0.5i)3-s + (2.72 + 1.57i)4-s + (−0.477 + 1.78i)5-s + (1.60 + 1.60i)6-s + (0.565 − 2.58i)7-s + (−1.83 − 1.83i)8-s + (0.499 + 0.866i)9-s + (2.09 − 3.62i)10-s + (−2.73 + 0.733i)11-s + (−1.57 − 2.72i)12-s + (−1.91 − 3.05i)13-s + (−2.75 + 5.33i)14-s + (1.30 − 1.30i)15-s + (−0.201 − 0.348i)16-s + (1.30 − 2.26i)17-s + ⋯
L(s)  = 1  + (−1.54 − 0.415i)2-s + (−0.499 − 0.288i)3-s + (1.36 + 0.786i)4-s + (−0.213 + 0.797i)5-s + (0.654 + 0.654i)6-s + (0.213 − 0.976i)7-s + (−0.648 − 0.648i)8-s + (0.166 + 0.288i)9-s + (0.662 − 1.14i)10-s + (−0.825 + 0.221i)11-s + (−0.453 − 0.786i)12-s + (−0.532 − 0.846i)13-s + (−0.736 + 1.42i)14-s + (0.337 − 0.337i)15-s + (−0.0503 − 0.0871i)16-s + (0.316 − 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.693 - 0.720i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.693 - 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0237756 + 0.0559178i\)
\(L(\frac12)\) \(\approx\) \(0.0237756 + 0.0559178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.565 + 2.58i)T \)
13 \( 1 + (1.91 + 3.05i)T \)
good2 \( 1 + (2.19 + 0.587i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.477 - 1.78i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.73 - 0.733i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.30 + 2.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.05 - 7.68i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (6.09 - 3.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.00T + 29T^{2} \)
31 \( 1 + (8.03 - 2.15i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.48 - 5.55i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.97 + 4.97i)T + 41iT^{2} \)
43 \( 1 - 5.75iT - 43T^{2} \)
47 \( 1 + (8.35 + 2.23i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.51 + 7.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.311 + 1.16i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.80 + 1.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.00997 + 0.0372i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.50 - 1.50i)T - 71iT^{2} \)
73 \( 1 + (0.758 + 2.83i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.66 - 2.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.45 - 5.45i)T + 83iT^{2} \)
89 \( 1 + (-3.01 - 0.808i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.47 + 7.47i)T + 97iT^{2} \)
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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

−11.83410594155456922094361279954, −10.97251479550612007093930835355, −10.24654539166802063812147406103, −9.899335100012893626883395904202, −8.123948124208389342593253428282, −7.66540307264745336480233413881, −6.86621959706493138651046386218, −5.31658980666386410796644885282, −3.41662005729882134788571955480, −1.79070049263412361975195769528, 0.07744566216712735892255253429, 2.10888352727641081992406134377, 4.53265748522472572990600228977, 5.65095312526839531104082359690, 6.80203634317896261169322475745, 7.962879904026894871147756516633, 8.802554734575643033625611001544, 9.342057224649906801619781864814, 10.44605524335851068685554666506, 11.27102580714473963247086131090

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