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Properties

Label 1-273-273.242-r0-0-0
Degree $1$
Conductor $273$
Sign $0.349 + 0.936i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (0.866 − 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (0.866 − 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.349 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.898030195 + 1.317295258i\)
\(L(\frac12)\) \(\approx\) \(1.898030195 + 1.317295258i\)
\(L(1)\) \(\approx\) \(1.718921942 + 0.7569514352i\)
\(L(1)\) \(\approx\) \(1.718921942 + 0.7569514352i\)

Euler product

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

−25.15929577659941670158077186026, −24.63945901852297801364352180874, −23.66384238374497181642996398101, −22.64914351897255866638569922265, −21.84512256893405610963135685573, −21.11436864056333468604979821610, −20.21445271835393410526363949035, −19.47322247435149370802082633636, −18.26512336659239815369669729514, −17.165655276987720880873107489547, −16.28868046864084652870402278093, −14.90807209651236337507769459089, −14.36596392464286858292501617751, −13.15949590118382373158030216137, −12.632321456603661489722909384993, −11.55800998283664896910639922310, −10.38409416953632651163662814742, −9.61969747522446328116479722078, −8.4415575152061840633210008116, −6.64651441161728679079050484816, −6.023433925372479159635151544283, −4.761715378040968313197858678066, −3.91581484910928015210052074736, −2.32555232006560521208377276346, −1.44815680842568669196518181619, 1.96223697104925034890612309862, 3.088409274252249262447154656200, 4.28489618161643468522049171341, 5.53707206997284632578239447032, 6.39415248053309884703327714406, 7.19666914184150368575570591791, 8.58494244674683300563381480293, 9.65858729171633752112231778210, 11.06117382358985424906276855102, 11.79073126046544048830562922507, 13.199401829271052854336583554261, 13.72079018379754602788880681554, 14.65471046304922140208226252619, 15.47800651309281819808603188359, 16.702714564985387829120435804089, 17.34790615088346621855655675605, 18.36422673125357417717716596564, 19.64341744088417422567679471163, 20.717874666194723125453596802916, 21.67893603859083278274066073828, 22.18296565159319623604921052358, 23.062101943205842737729939233908, 24.18997364710749574652796349579, 24.89507906088240984610549672591, 25.7094905529654810775386008991

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