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Properties

Label 2-273-91.74-c1-0-13
Degree $2$
Conductor $273$
Sign $0.991 + 0.130i$
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  + 2.62·2-s + (−0.5 − 0.866i)3-s + 4.89·4-s + (0.734 + 1.27i)5-s + (−1.31 − 2.27i)6-s + (−2.15 + 1.53i)7-s + 7.59·8-s + (−0.499 + 0.866i)9-s + (1.92 + 3.34i)10-s + (−2.24 − 3.88i)11-s + (−2.44 − 4.23i)12-s + (−3.58 + 0.374i)13-s + (−5.65 + 4.03i)14-s + (0.734 − 1.27i)15-s + 10.1·16-s − 3.62·17-s + ⋯
L(s)  = 1  + 1.85·2-s + (−0.288 − 0.499i)3-s + 2.44·4-s + (0.328 + 0.569i)5-s + (−0.535 − 0.928i)6-s + (−0.813 + 0.581i)7-s + 2.68·8-s + (−0.166 + 0.288i)9-s + (0.609 + 1.05i)10-s + (−0.675 − 1.17i)11-s + (−0.706 − 1.22i)12-s + (−0.994 + 0.103i)13-s + (−1.51 + 1.07i)14-s + (0.189 − 0.328i)15-s + 2.54·16-s − 0.880·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05564 - 0.200360i\)
\(L(\frac12)\) \(\approx\) \(3.05564 - 0.200360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 1.53i)T \)
13 \( 1 + (3.58 - 0.374i)T \)
good2 \( 1 - 2.62T + 2T^{2} \)
5 \( 1 + (-0.734 - 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.24 + 3.88i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 + (-1.50 + 2.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.74T + 23T^{2} \)
29 \( 1 + (3.98 - 6.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.552 - 0.957i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.15T + 37T^{2} \)
41 \( 1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.54 - 4.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.69 - 2.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.25 - 7.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 + (3.81 - 6.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.33 + 4.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.24 - 2.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.03 + 5.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.59 + 6.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.30T + 83T^{2} \)
89 \( 1 - 6.36T + 89T^{2} \)
97 \( 1 + (4.02 + 6.96i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.22320629386568903296372325629, −11.18266530811918133864228665653, −10.63523275325256999147990190046, −9.010027409267881179028072077612, −7.26926824562592672185248956254, −6.63546538866315430189687305206, −5.71667296787273648592545740290, −4.91514141234769563501323197835, −3.14596193241913266160190385908, −2.53130577281602420829985116389, 2.39768820835365890789084487643, 3.76484635422516065853440625903, 4.80730483301816440762667431204, 5.39793721856737542266062785224, 6.69203984794969310423427550500, 7.44377062540387762860282390442, 9.434642585179533179349968238298, 10.27054153574876370347316068636, 11.24783601410585626379891197568, 12.30651549272108719093651156896

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