Properties

 Label 1-153-153.115-r0-0-0 Degree $1$ Conductor $153$ Sign $-0.978 - 0.208i$ Analytic cond. $0.710529$ Root an. cond. $0.710529$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Learn more

Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)7-s − 8-s − i·10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)14-s + (−0.5 − 0.866i)16-s − 19-s + (0.866 − 0.5i)20-s + (−0.866 − 0.5i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)25-s − 26-s + ⋯
 L(s)  = 1 + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)7-s − 8-s − i·10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)14-s + (−0.5 − 0.866i)16-s − 19-s + (0.866 − 0.5i)20-s + (−0.866 − 0.5i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)25-s − 26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$153$$    =    $$3^{2} \cdot 17$$ Sign: $-0.978 - 0.208i$ Analytic conductor: $$0.710529$$ Root analytic conductor: $$0.710529$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{153} (115, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 153,\ (0:\ ),\ -0.978 - 0.208i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-0.05507036796 + 0.5231636797i$$ $$L(\frac12)$$ $$\approx$$ $$-0.05507036796 + 0.5231636797i$$ $$L(1)$$ $$\approx$$ $$0.5870585671 + 0.4918144031i$$ $$L(1)$$ $$\approx$$ $$0.5870585671 + 0.4918144031i$$

Euler product

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

Imaginary part of the first few zeros on the critical line

−27.42713838184377814267701506314, −26.85696723409628489507703474709, −25.67792760057699846960918793983, −24.14227508036080776917959417839, −23.241467684020296047203010874864, −22.69250427395148705707578450497, −21.64327252166015690256704084692, −20.494429629945492776157901951342, −19.517045959260959995603027424079, −19.00296148800790753654699080678, −17.81360903343362395272325739294, −16.21455949080650695757867097274, −15.251091966059990108377199502582, −14.2420311687066300310323764585, −12.95405379610095804613089373285, −12.37871183840634224290148943203, −10.71850420557660715859166582940, −10.58561122877194652088898342901, −8.96221495820157602615499777513, −7.52340723102066813673355795443, −6.20048796173441860801425114668, −4.76599089137170499734677993058, −3.48981452705039028484394970190, −2.69277655058631903020540283635, −0.36428106731772197702084654273, 2.75523777574550349765606232706, 4.144546991300662999821083577167, 5.1124376602232793935178709655, 6.48501244834262990122613008180, 7.52132822093107052023961356403, 8.6088198242406346945349138705, 9.64374810645779449747304705460, 11.51357766726826334071932987885, 12.59145764404587069214299162702, 13.14935683956751780487129642214, 14.7232994229393485399754874718, 15.5285828186028154815747150849, 16.2948639284151775985769074737, 17.21593135738799791423412437153, 18.62334610226126161378980610052, 19.50267011502027088218274510203, 20.8970339498399154508338490298, 21.805748989894865571368973322502, 23.01901507596655775051762678030, 23.53006604794989355741548802114, 24.54973094856661349500719383311, 25.55315605556584154783620514643, 26.352740042027099656699520505580, 27.39101313246337259117729402027, 28.43168392189169240456430711870