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Properties

Label 2-273-91.73-c1-0-9
Degree $2$
Conductor $273$
Sign $0.285 - 0.958i$
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  + (−0.164 + 0.612i)2-s + (0.866 + 0.5i)3-s + (1.38 + 0.798i)4-s + (2.57 + 0.690i)5-s + (−0.448 + 0.448i)6-s + (−1.84 + 1.89i)7-s + (−1.61 + 1.61i)8-s + (0.499 + 0.866i)9-s + (−0.845 + 1.46i)10-s + (−1.15 − 4.30i)11-s + (0.798 + 1.38i)12-s + (−2.17 − 2.87i)13-s + (−0.860 − 1.44i)14-s + (1.88 + 1.88i)15-s + (0.874 + 1.51i)16-s + (2.00 − 3.47i)17-s + ⋯
L(s)  = 1  + (−0.116 + 0.433i)2-s + (0.499 + 0.288i)3-s + (0.691 + 0.399i)4-s + (1.15 + 0.308i)5-s + (−0.183 + 0.183i)6-s + (−0.696 + 0.717i)7-s + (−0.570 + 0.570i)8-s + (0.166 + 0.288i)9-s + (−0.267 + 0.462i)10-s + (−0.347 − 1.29i)11-s + (0.230 + 0.399i)12-s + (−0.603 − 0.797i)13-s + (−0.229 − 0.384i)14-s + (0.486 + 0.486i)15-s + (0.218 + 0.378i)16-s + (0.486 − 0.843i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.285 - 0.958i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38075 + 1.02888i\)
\(L(\frac12)\) \(\approx\) \(1.38075 + 1.02888i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 - 1.89i)T \)
13 \( 1 + (2.17 + 2.87i)T \)
good2 \( 1 + (0.164 - 0.612i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-2.57 - 0.690i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.15 + 4.30i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.00 + 3.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.23 + 1.13i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.04 + 1.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.379T + 29T^{2} \)
31 \( 1 + (-2.71 - 10.1i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.246 - 0.0661i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.78 + 7.78i)T - 41iT^{2} \)
43 \( 1 - 3.61iT - 43T^{2} \)
47 \( 1 + (-1.12 + 4.20i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.54 - 9.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.44 + 0.924i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-12.0 + 6.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.6 - 3.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.43 + 5.43i)T + 71iT^{2} \)
73 \( 1 + (7.80 - 2.09i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.48 - 2.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.78 - 8.78i)T - 83iT^{2} \)
89 \( 1 + (0.166 - 0.621i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.39 + 7.39i)T - 97iT^{2} \)
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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

−12.24058168422505401912938828168, −10.92890849955049752558806851809, −10.18152926122629398061308697011, −9.076280958048139847649274050822, −8.372474972082871698801993529086, −7.10874497343836281633993444250, −6.11677300472363617108023665203, −5.35827607560390113536705102180, −3.05913224529341116025508139488, −2.57402634155542420880073152675, 1.62530218916128455181495263095, 2.56091253920565183341579681469, 4.25712147118553793232341432315, 5.89349490593356239619607847950, 6.73610285222994153284625496407, 7.65757070923615643663958201377, 9.321103894314704252528141753779, 9.839935533814534306479463226505, 10.44487967096819614277990035929, 11.77249500494272126597070618420

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