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Properties

Label 2-273-39.8-c1-0-16
Degree $2$
Conductor $273$
Sign $0.990 - 0.136i$
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  + (−1.48 + 1.48i)2-s + (1.53 − 0.809i)3-s − 2.42i·4-s + (0.854 − 0.854i)5-s + (−1.07 + 3.48i)6-s + (0.707 − 0.707i)7-s + (0.634 + 0.634i)8-s + (1.69 − 2.47i)9-s + 2.54i·10-s + (0.183 + 0.183i)11-s + (−1.96 − 3.71i)12-s + (−2.40 − 2.68i)13-s + 2.10i·14-s + (0.616 − 1.99i)15-s + 2.96·16-s + 7.60·17-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)2-s + (0.884 − 0.467i)3-s − 1.21i·4-s + (0.382 − 0.382i)5-s + (−0.438 + 1.42i)6-s + (0.267 − 0.267i)7-s + (0.224 + 0.224i)8-s + (0.563 − 0.826i)9-s + 0.803i·10-s + (0.0554 + 0.0554i)11-s + (−0.566 − 1.07i)12-s + (−0.666 − 0.745i)13-s + 0.562i·14-s + (0.159 − 0.516i)15-s + 0.741·16-s + 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05919 + 0.0725027i\)
\(L(\frac12)\) \(\approx\) \(1.05919 + 0.0725027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 + 0.809i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (2.40 + 2.68i)T \)
good2 \( 1 + (1.48 - 1.48i)T - 2iT^{2} \)
5 \( 1 + (-0.854 + 0.854i)T - 5iT^{2} \)
11 \( 1 + (-0.183 - 0.183i)T + 11iT^{2} \)
17 \( 1 - 7.60T + 17T^{2} \)
19 \( 1 + (4.33 + 4.33i)T + 19iT^{2} \)
23 \( 1 + 0.600T + 23T^{2} \)
29 \( 1 - 2.36iT - 29T^{2} \)
31 \( 1 + (-6.76 - 6.76i)T + 31iT^{2} \)
37 \( 1 + (2.20 - 2.20i)T - 37iT^{2} \)
41 \( 1 + (2.79 - 2.79i)T - 41iT^{2} \)
43 \( 1 - 2.48iT - 43T^{2} \)
47 \( 1 + (-5.78 - 5.78i)T + 47iT^{2} \)
53 \( 1 + 5.27iT - 53T^{2} \)
59 \( 1 + (7.52 + 7.52i)T + 59iT^{2} \)
61 \( 1 + 0.457T + 61T^{2} \)
67 \( 1 + (5.51 + 5.51i)T + 67iT^{2} \)
71 \( 1 + (9.60 - 9.60i)T - 71iT^{2} \)
73 \( 1 + (-4.31 + 4.31i)T - 73iT^{2} \)
79 \( 1 + 4.20T + 79T^{2} \)
83 \( 1 + (-0.439 + 0.439i)T - 83iT^{2} \)
89 \( 1 + (-3.82 - 3.82i)T + 89iT^{2} \)
97 \( 1 + (-5.33 - 5.33i)T + 97iT^{2} \)
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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

−12.17752748239052500710399826222, −10.40935403413573543697043238269, −9.689042064498282408077976126119, −8.816260767495466149276719366245, −8.022995495385050677840756334997, −7.34898202129635595628194886131, −6.35948109583367870695173108178, −5.05191867351370613656469273637, −3.13834340183467718195806597135, −1.20014890213784705405112039147, 1.85042212795959648786650221064, 2.80966217214285818213056190239, 4.13319121971890160119135389773, 5.82165755898837503677698870976, 7.56285675652852175016402606786, 8.343367954912670502384610900574, 9.241430511858722805944958351839, 10.15970504627019651135505472834, 10.37439520376207919891942111787, 11.79045124514921740281697541060

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