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Properties

Label 2-273-21.20-c1-0-23
Degree $2$
Conductor $273$
Sign $-0.0764 + 0.997i$
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  − 1.71i·2-s + (1.66 + 0.466i)3-s − 0.943·4-s − 2.59·5-s + (0.800 − 2.86i)6-s + (2.59 − 0.515i)7-s − 1.81i·8-s + (2.56 + 1.55i)9-s + 4.45i·10-s − 3.05i·11-s + (−1.57 − 0.440i)12-s − i·13-s + (−0.885 − 4.45i)14-s + (−4.32 − 1.21i)15-s − 4.99·16-s + 2.57·17-s + ⋯
L(s)  = 1  − 1.21i·2-s + (0.963 + 0.269i)3-s − 0.471·4-s − 1.16·5-s + (0.326 − 1.16i)6-s + (0.980 − 0.195i)7-s − 0.640i·8-s + (0.854 + 0.518i)9-s + 1.40i·10-s − 0.920i·11-s + (−0.454 − 0.127i)12-s − 0.277i·13-s + (−0.236 − 1.18i)14-s + (−1.11 − 0.312i)15-s − 1.24·16-s + 0.625·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0764 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0764 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0764 + 0.997i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0764 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12515 - 1.21467i\)
\(L(\frac12)\) \(\approx\) \(1.12515 - 1.21467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 - 0.466i)T \)
7 \( 1 + (-2.59 + 0.515i)T \)
13 \( 1 + iT \)
good2 \( 1 + 1.71iT - 2T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 + 3.05iT - 11T^{2} \)
17 \( 1 - 2.57T + 17T^{2} \)
19 \( 1 + 0.733iT - 19T^{2} \)
23 \( 1 - 5.99iT - 23T^{2} \)
29 \( 1 - 0.420iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 2.63T + 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 - 5.06T + 59T^{2} \)
61 \( 1 + 9.31iT - 61T^{2} \)
67 \( 1 - 2.59T + 67T^{2} \)
71 \( 1 - 4.49iT - 71T^{2} \)
73 \( 1 + 3.93iT - 73T^{2} \)
79 \( 1 - 8.55T + 79T^{2} \)
83 \( 1 + 1.25T + 83T^{2} \)
89 \( 1 + 7.71T + 89T^{2} \)
97 \( 1 + 8.62iT - 97T^{2} \)
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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

−11.55880379981277730531360805733, −10.84913580023548743006257834061, −10.01099191776776222493953244263, −8.773281411707601746613687628345, −8.033843205558254500995591054927, −7.12787346202390952905003354658, −4.98494694236827254330133969505, −3.75482817753452352196077295269, −3.14584029501722412778140345653, −1.44697504141720429750149696020, 2.18419403715004224385854230681, 3.99692611398801193274300590219, 4.98013831836382046451688162448, 6.58677176733139061059780798202, 7.49814984393221862974600004835, 8.046716903192691477684974680485, 8.675747611365086201848274944320, 10.01600183560135879659426025542, 11.49157165764350777233509978312, 12.08852980351784571091523226812

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