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Properties

Label 1-456-456.59-r1-0-0
Degree $1$
Conductor $456$
Sign $0.624 - 0.780i$
Analytic cond. $49.0040$
Root an. cond. $49.0040$
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  + (0.766 − 0.642i)5-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (0.939 + 0.342i)17-s + (0.766 + 0.642i)23-s + (0.173 − 0.984i)25-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)35-s + 37-s + (0.173 + 0.984i)41-s + (0.766 − 0.642i)43-s + (−0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (0.939 + 0.342i)17-s + (0.766 + 0.642i)23-s + (0.173 − 0.984i)25-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)35-s + 37-s + (0.173 + 0.984i)41-s + (0.766 − 0.642i)43-s + (−0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(49.0040\)
Root analytic conductor: \(49.0040\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 456,\ (1:\ ),\ 0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.585469377 - 1.242285378i\)
\(L(\frac12)\) \(\approx\) \(2.585469377 - 1.242285378i\)
\(L(1)\) \(\approx\) \(1.437608009 - 0.3193225139i\)
\(L(1)\) \(\approx\) \(1.437608009 - 0.3193225139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−23.980305758281668802241377024535, −22.89446726100419590987961201604, −21.967750815137690273835016223496, −21.41082174578430940850424924306, −20.76155405717243149418736910108, −19.26054040797917587179315400112, −18.70357052757350882720055272181, −18.008902505926892587996031340036, −16.9387376313041601658496121918, −16.20946459288945596562520505251, −14.90799965501601132714415518683, −14.34209055230083683500901356381, −13.59570772250717125335676058416, −12.35589354512300621196696785702, −11.44384433676137165916262325976, −10.72531776619921729041921963655, −9.45245496489848158430624459810, −8.88810707536793821046229276655, −7.67305918682382145660996543794, −6.45018385326706630807127641447, −5.83241472749000136362849960386, −4.70949986992775859652919264453, −3.27733056413544205841788283486, −2.33259109268789141497548419386, −1.17679612168490789566685988397, 0.89857233097037794163865410333, 1.656704100301222934012693557498, 3.18361376016623147993850132826, 4.45586923674436124103981090873, 5.24104426131829081267178013531, 6.33719968759698803370856219553, 7.49279825746591799351630008477, 8.32429535132763682486834149464, 9.5382301145354938301041842214, 10.17272339883044938153456022148, 11.17063804543025633644352789947, 12.42445010899439557711844593643, 13.03578701473699624956739309532, 14.06376555522987807962108612413, 14.74474323340437179184081325007, 15.91234741445661795915724908479, 16.998782716366493564137077299681, 17.42918830261784495740188742452, 18.20034450126631835136019433554, 19.62230510717824834877062890513, 20.243047185391857943715296180124, 21.01005621404994517694351890357, 21.7366258993583810267911517259, 23.00313694909036347424327275094, 23.457542673819785939595125617758

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