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Properties

Label 1-153-153.4-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 $0$

Origins

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Normalization:  

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: $\chi_{153} (4, \cdot )$
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 \( 1 + (0.866 + 0.5i)T \)
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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

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

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