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Properties

Label 1-1840-1840.619-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.0298 + 0.999i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.281 + 0.959i)3-s + (−0.142 − 0.989i)7-s + (−0.841 + 0.540i)9-s + (0.909 + 0.415i)11-s + (−0.989 − 0.142i)13-s + (−0.654 − 0.755i)17-s + (0.755 + 0.654i)19-s + (0.909 − 0.415i)21-s + (−0.755 − 0.654i)27-s + (−0.755 + 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + (0.540 + 0.841i)37-s + (−0.142 − 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)3-s + (−0.142 − 0.989i)7-s + (−0.841 + 0.540i)9-s + (0.909 + 0.415i)11-s + (−0.989 − 0.142i)13-s + (−0.654 − 0.755i)17-s + (0.755 + 0.654i)19-s + (0.909 − 0.415i)21-s + (−0.755 − 0.654i)27-s + (−0.755 + 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + (0.540 + 0.841i)37-s + (−0.142 − 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0298 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0298 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.0298 + 0.999i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.0298 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060108183 + 1.028933521i\)
\(L(\frac12)\) \(\approx\) \(1.060108183 + 1.028933521i\)
\(L(1)\) \(\approx\) \(1.039306921 + 0.3495547619i\)
\(L(1)\) \(\approx\) \(1.039306921 + 0.3495547619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (0.909 + 0.415i)T \)
13 \( 1 + (-0.989 - 0.142i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + (-0.755 + 0.654i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.540 + 0.841i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + T \)
53 \( 1 + (0.989 - 0.142i)T \)
59 \( 1 + (0.989 + 0.142i)T \)
61 \( 1 + (0.281 - 0.959i)T \)
67 \( 1 + (-0.909 + 0.415i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−19.64349947195756265806285231303, −19.327958244633279462932857135201, −18.61542135402433111861145531277, −17.7741742680904764581753381263, −17.23610629418057813546231133579, −16.41148513418964701982057888781, −15.212685918245873273635612339860, −14.916888919774450505718458996727, −13.930556581154834205891543451894, −13.312241483442772830786725589360, −12.47983078301594395616549969008, −11.799235153493148878804493117, −11.399543039531575737942104671451, −10.06911784194712529079521705948, −9.046570895860533310374079463728, −8.81370923659284247357188098187, −7.733781862003879065673323239943, −7.016547375256140900109124932177, −6.17538169357962723029441660180, −5.60956928577009929681770749393, −4.432803345043367634425889941, −3.34435782731343914944369584587, −2.45542824875883827684328399720, −1.83897872680240407901535011802, −0.56696979361272932322536924342, 1.0254636310617265097493722999, 2.32614052360574758264567253406, 3.26042989728627648152040086717, 4.06524815583852327344809506730, 4.69500458208780233061295902608, 5.50726371988532204385453878157, 6.73620025724582435991133491938, 7.34764329761228872630373243362, 8.27794627518638221571929848719, 9.25374150187365121498031240802, 9.82748002317426407686878084672, 10.35365891573178907273611222418, 11.35873985831430645184996959968, 11.928363383587875438301601231282, 13.02683165047019856764703397931, 13.94404568372536320406704068005, 14.36850965157621779802207531921, 15.1444175260757924138978637961, 15.93955612872897802217892276379, 16.72488632662965159361196734827, 17.13557309604823884819118724037, 17.954878331934211936852510712325, 19.09334327703563357159531817520, 19.90826370520777633008474036651, 20.23434114816654622881358254185

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