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Properties

Label 2-273-13.4-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.953 + 0.301i$
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  + (−0.654 − 0.378i)2-s + (0.5 − 0.866i)3-s + (−0.714 − 1.23i)4-s − 4.01i·5-s + (−0.654 + 0.378i)6-s + (−0.866 + 0.5i)7-s + 2.59i·8-s + (−0.499 − 0.866i)9-s + (−1.51 + 2.62i)10-s + (−0.588 − 0.339i)11-s − 1.42·12-s + (−0.823 + 3.51i)13-s + 0.756·14-s + (−3.47 − 2.00i)15-s + (−0.448 + 0.776i)16-s + (1.48 + 2.57i)17-s + ⋯
L(s)  = 1  + (−0.463 − 0.267i)2-s + (0.288 − 0.499i)3-s + (−0.357 − 0.618i)4-s − 1.79i·5-s + (−0.267 + 0.154i)6-s + (−0.327 + 0.188i)7-s + 0.916i·8-s + (−0.166 − 0.288i)9-s + (−0.480 + 0.831i)10-s + (−0.177 − 0.102i)11-s − 0.412·12-s + (−0.228 + 0.973i)13-s + 0.202·14-s + (−0.897 − 0.518i)15-s + (−0.112 + 0.194i)16-s + (0.360 + 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.953 + 0.301i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.953 + 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125616 - 0.814254i\)
\(L(\frac12)\) \(\approx\) \(0.125616 - 0.814254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.823 - 3.51i)T \)
good2 \( 1 + (0.654 + 0.378i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 4.01iT - 5T^{2} \)
11 \( 1 + (0.588 + 0.339i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.48 - 2.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.795 + 0.459i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.29 + 7.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.98 + 6.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.29iT - 31T^{2} \)
37 \( 1 + (4.70 + 2.71i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.98 + 4.61i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.53 - 4.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.03iT - 47T^{2} \)
53 \( 1 - 1.32T + 53T^{2} \)
59 \( 1 + (-4.18 + 2.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.49 - 2.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.08 - 1.20i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.82iT - 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 4.78iT - 83T^{2} \)
89 \( 1 + (3.91 + 2.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.40 - 4.27i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.65838200978368841316508283011, −10.28934386148160839397427392761, −9.312050022033317065288525616496, −8.768862329308439238533781220994, −8.033126550983874285033163640240, −6.39248793523794988539803803881, −5.26808623040620186370076105468, −4.31945881820224903059872163924, −2.08243932938403338869906734902, −0.73181851413728863054185296625, 3.06672338010932645595048719475, 3.43416963417695317603139617240, 5.24516321808207292013407986433, 6.87912257830600852007366033721, 7.35307767554497246008831406337, 8.420818213170197494998991321245, 9.680982368568664995008735312270, 10.20816572870266246988894815335, 11.11874097050881289426002224701, 12.25682342013736928633250058430

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