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Properties

Label 1-273-273.74-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.981 - 0.190i$
Analytic cond. $29.3379$
Root an. cond. $29.3379$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 16-s − 17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 16-s − 17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.981 - 0.190i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05596830829 - 0.5811735144i\)
\(L(\frac12)\) \(\approx\) \(0.05596830829 - 0.5811735144i\)
\(L(1)\) \(\approx\) \(0.6227778316 - 0.2144214012i\)
\(L(1)\) \(\approx\) \(0.6227778316 - 0.2144214012i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.862929743784230332465419300341, −25.33784959892097278091778028329, −24.4194992893171688906416574502, −23.18216604399375264287293132067, −22.179128796667823236596917930823, −21.2683985992281807527473330, −20.21929104932427196391759766951, −19.42498826894750866698197990891, −18.40649948081991806861162836276, −17.74324900548110056425780447305, −17.00000914550441480059369901024, −15.775065749557453157839597179708, −14.9229349130929600486381779089, −14.02414092958712568202381988535, −12.55836660962568038906668819356, −11.5453977036981293487743060311, −10.51275600528565966119251985201, −9.84724170866970773820516948778, −8.83092607476808127635986294516, −7.60884957007114834297952552670, −6.70006704264767805235981649736, −5.89171218002276868105923733817, −4.0311766813262257963901828446, −2.5361218734772927818316866765, −1.67776912565855837974280303802, 0.24122640639059654399964346160, 1.44766661926720139169172158397, 2.66207672262993431778816640029, 4.320291957795171835338925108750, 5.80189142770056707891918001227, 6.61956181816379746364276431945, 8.03520615617222455689369618492, 8.871855127644025403799387120086, 9.53686987859165478609346779462, 10.76806211715859308193448597842, 11.63088092534190044772193734452, 12.759874838867596075151135457, 13.77658138926850863926348028935, 15.08166779295204509690437827036, 16.17112915455972105928364797532, 16.78062361480597135622600622336, 17.68775574912246694166718245986, 18.473758670974486290883561995128, 19.85700950776515259411936029047, 20.02698683031440979449950626695, 21.43114326424526832752134561277, 21.89091280966704565256563285380, 23.70166428234352515109554532730, 24.37544866307906183691575350681, 25.082975802395893728946303025809

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