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Properties

Label 1-2151-2151.122-r1-0-0
Degree $1$
Conductor $2151$
Sign $0.159 - 0.987i$
Analytic cond. $231.157$
Root an. cond. $231.157$
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  + (0.307 − 0.951i)2-s + (−0.811 − 0.584i)4-s + (0.859 − 0.511i)5-s + (−0.657 − 0.753i)7-s + (−0.805 + 0.592i)8-s + (−0.222 − 0.974i)10-s + (0.963 + 0.269i)11-s + (−0.453 − 0.891i)13-s + (−0.919 + 0.393i)14-s + (0.315 + 0.948i)16-s + (0.118 + 0.992i)17-s + (0.773 − 0.633i)19-s + (−0.996 − 0.0878i)20-s + (0.552 − 0.833i)22-s + (0.153 + 0.988i)23-s + ⋯
L(s)  = 1  + (0.307 − 0.951i)2-s + (−0.811 − 0.584i)4-s + (0.859 − 0.511i)5-s + (−0.657 − 0.753i)7-s + (−0.805 + 0.592i)8-s + (−0.222 − 0.974i)10-s + (0.963 + 0.269i)11-s + (−0.453 − 0.891i)13-s + (−0.919 + 0.393i)14-s + (0.315 + 0.948i)16-s + (0.118 + 0.992i)17-s + (0.773 − 0.633i)19-s + (−0.996 − 0.0878i)20-s + (0.552 − 0.833i)22-s + (0.153 + 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(231.157\)
Root analytic conductor: \(231.157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2151,\ (1:\ ),\ 0.159 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.090299531 - 1.780373251i\)
\(L(\frac12)\) \(\approx\) \(2.090299531 - 1.780373251i\)
\(L(1)\) \(\approx\) \(1.074122285 - 0.7712760789i\)
\(L(1)\) \(\approx\) \(1.074122285 - 0.7712760789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 \)
good2 \( 1 + (0.307 - 0.951i)T \)
5 \( 1 + (0.859 - 0.511i)T \)
7 \( 1 + (-0.657 - 0.753i)T \)
11 \( 1 + (0.963 + 0.269i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (0.118 + 0.992i)T \)
19 \( 1 + (0.773 - 0.633i)T \)
23 \( 1 + (0.153 + 0.988i)T \)
29 \( 1 + (0.0835 + 0.996i)T \)
31 \( 1 + (-0.967 - 0.252i)T \)
37 \( 1 + (-0.720 + 0.692i)T \)
41 \( 1 + (0.744 + 0.667i)T \)
43 \( 1 + (0.689 - 0.724i)T \)
47 \( 1 + (0.993 + 0.114i)T \)
53 \( 1 + (-0.144 - 0.989i)T \)
59 \( 1 + (-0.727 - 0.686i)T \)
61 \( 1 + (0.595 + 0.803i)T \)
67 \( 1 + (0.650 + 0.759i)T \)
71 \( 1 + (0.850 + 0.526i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (0.854 - 0.518i)T \)
83 \( 1 + (0.530 + 0.847i)T \)
89 \( 1 + (-0.581 + 0.813i)T \)
97 \( 1 + (0.762 - 0.647i)T \)
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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

−19.53863727488651343635169899465, −18.69651541989212087979746610528, −18.39490946666630669800329263530, −17.46106558058261450515062684109, −16.75362680218275309250306546505, −16.22385963686382895000129921253, −15.44315549702273139552608604358, −14.45546678075047952781553498026, −14.17754570910658255728602726275, −13.53374773943465362935984528112, −12.41115062892919284500892415728, −12.08286580659521580014368086275, −10.96475045338603114987561554452, −9.72456379055232908319024805033, −9.32861109584601846288326027573, −8.81710006045898973455057934644, −7.51144592150169603234771898686, −6.90357021690773565475097685693, −6.15261926427254533626269618417, −5.690443476745307470396573164041, −4.74331099308230482592436890448, −3.734669536351397320811290658161, −2.90641295165293576350837474885, −2.02469845057178623222822789984, −0.56073804473185966205083593980, 0.78801143670188646449507536555, 1.323809469657984843427910402540, 2.332199138461453773840286509457, 3.35905466452148803418421727392, 3.940308267702818670789489409190, 5.003392242817738003502026970863, 5.59890687349693456029744186054, 6.48759101023558053852359697849, 7.42085769581558201924405700675, 8.61983519655311002958348591064, 9.38010467798986857509470079809, 9.84645790289119329263135418391, 10.54283585902642934951478488114, 11.31728198419064994358122175725, 12.407785100157891874109834653190, 12.75969544336458609991208328711, 13.481673918106065712597173043254, 14.12099595069455833755238187049, 14.82145990691993975439784872510, 15.79155238588764776199589635680, 16.85703565103450284048541309410, 17.42016095335599109817501459191, 17.86092932374239095206499595690, 18.98036673254049159880325518800, 19.74945974590467907778172960624

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