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Properties

Label 1-847-847.19-r0-0-0
Degree $1$
Conductor $847$
Sign $0.992 + 0.124i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.879 + 0.475i)2-s + (0.104 + 0.994i)3-s + (0.548 − 0.836i)4-s + (0.999 + 0.0190i)5-s + (−0.564 − 0.825i)6-s + (−0.0855 + 0.996i)8-s + (−0.978 + 0.207i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (−0.398 − 0.917i)16-s + (0.345 − 0.938i)17-s + (0.761 − 0.647i)18-s + (0.272 − 0.962i)19-s + (0.564 − 0.825i)20-s + ⋯
L(s)  = 1  + (−0.879 + 0.475i)2-s + (0.104 + 0.994i)3-s + (0.548 − 0.836i)4-s + (0.999 + 0.0190i)5-s + (−0.564 − 0.825i)6-s + (−0.0855 + 0.996i)8-s + (−0.978 + 0.207i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (−0.398 − 0.917i)16-s + (0.345 − 0.938i)17-s + (0.761 − 0.647i)18-s + (0.272 − 0.962i)19-s + (0.564 − 0.825i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.992 + 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043841059 + 0.06537559544i\)
\(L(\frac12)\) \(\approx\) \(1.043841059 + 0.06537559544i\)
\(L(1)\) \(\approx\) \(0.8137516821 + 0.2309987378i\)
\(L(1)\) \(\approx\) \(0.8137516821 + 0.2309987378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.879 + 0.475i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (0.999 + 0.0190i)T \)
13 \( 1 + (-0.254 - 0.967i)T \)
17 \( 1 + (0.345 - 0.938i)T \)
19 \( 1 + (0.272 - 0.962i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (-0.161 - 0.986i)T \)
37 \( 1 + (-0.625 + 0.780i)T \)
41 \( 1 + (0.610 - 0.791i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.761 + 0.647i)T \)
53 \( 1 + (-0.398 + 0.917i)T \)
59 \( 1 + (0.991 - 0.132i)T \)
61 \( 1 + (-0.851 + 0.524i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (0.00951 - 0.999i)T \)
79 \( 1 + (-0.797 - 0.603i)T \)
83 \( 1 + (-0.736 + 0.676i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.516 - 0.856i)T \)
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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

−21.749046386588719623178815125277, −21.26980915682474233243759949298, −20.29739395669826600171744603684, −19.56823542116954956051217705643, −18.83207840841162452464472412578, −18.13101033595931627158604567337, −17.55949009916603852799722145796, −16.782966369527719566026359959472, −16.09662387165802443972044713401, −14.440566739584411805602373807596, −14.072543874754640874261474589256, −12.81158679543475445385334307779, −12.446026129207409268169922426966, −11.505156748128558814329071322149, −10.503264782344258760456532058731, −9.74046386085206942244626271959, −8.82875892708880804198735547838, −8.15609544608684983578253956930, −7.12117715392593038439144754806, −6.42384541046793405704143176636, −5.55112874086227864771288952584, −3.89206588156969133018527798882, −2.74338922316065456868366053399, −1.83297308905320809882023276458, −1.2905579547060290936946946150, 0.64426359067210974914306842944, 2.24956276691248895247300377676, 2.955793818760862980928983297827, 4.56191105695859380709588393353, 5.49004883298911545582981183242, 5.99995412299719804409295163751, 7.247944620058132554674686489107, 8.173231238815305906601715609629, 9.17056483063883293325949939243, 9.67824538477785960910329053320, 10.35457561305889482677235146949, 11.090518847381730582619724948260, 12.146573111785771198489963225, 13.69997156114228032478289346199, 14.10343637480839406703062045394, 15.23996891861140562827802400619, 15.66708309784470271195043613272, 16.614077503552304582901685018464, 17.364396274121981769846489962045, 17.82765802856863431824133354665, 18.80960397994831031585401524529, 19.86117803522854484093134998977, 20.52067286239554990332928260973, 21.08572038002753743889355940826, 22.25142144792901363651413103689

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