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Properties

Label 1-847-847.285-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.817 - 0.575i$
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  + (0.786 − 0.618i)2-s + (0.5 − 0.866i)3-s + (0.235 − 0.971i)4-s + (0.995 − 0.0950i)5-s + (−0.142 − 0.989i)6-s + (−0.415 − 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.723 − 0.690i)10-s + (−0.723 − 0.690i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)15-s + (−0.888 − 0.458i)16-s + (0.981 − 0.189i)17-s + (−0.928 − 0.371i)18-s + (0.981 + 0.189i)19-s + (0.142 − 0.989i)20-s + ⋯
L(s)  = 1  + (0.786 − 0.618i)2-s + (0.5 − 0.866i)3-s + (0.235 − 0.971i)4-s + (0.995 − 0.0950i)5-s + (−0.142 − 0.989i)6-s + (−0.415 − 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.723 − 0.690i)10-s + (−0.723 − 0.690i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)15-s + (−0.888 − 0.458i)16-s + (0.981 − 0.189i)17-s + (−0.928 − 0.371i)18-s + (0.981 + 0.189i)19-s + (0.142 − 0.989i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.817 - 0.575i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.817 - 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9208078849 - 2.908527502i\)
\(L(\frac12)\) \(\approx\) \(0.9208078849 - 2.908527502i\)
\(L(1)\) \(\approx\) \(1.447600989 - 1.484913718i\)
\(L(1)\) \(\approx\) \(1.447600989 - 1.484913718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.786 - 0.618i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.995 - 0.0950i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (0.981 - 0.189i)T \)
19 \( 1 + (0.981 + 0.189i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.723 + 0.690i)T \)
37 \( 1 + (0.235 + 0.971i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.928 + 0.371i)T \)
53 \( 1 + (-0.888 + 0.458i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.0475 + 0.998i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−22.07993863278501769324928115314, −21.88808624067940323607424014119, −21.06410457523647181874185004216, −20.34266773711568336251181179283, −19.55631170279585228642087892030, −18.11452715075605145274551925101, −17.467731825143305998642449588573, −16.458832092846637756754395657256, −16.134544326210608174263749339740, −14.792188746456920896003215951766, −14.6017319447622230438082829327, −13.75658540484934482832855445152, −12.999701516246732246326476659131, −12.01071472140036659482295444647, −10.96960791182029993016680832303, −9.8585109438641985150565542592, −9.42583755739979567393887334390, −8.185636940153668148176069121931, −7.49889861662805630148350456686, −6.3083557244756864689769333127, −5.336386802375292309730849856823, −4.93961129409950535813614762763, −3.64055920089947915367887376588, −2.927036428885642803274960534793, −1.95027517455941843386528342380, 0.986828134031096991133255743492, 1.93547099510836845798228752417, 2.66085757369483718239616566082, 3.57749549521923992331635034957, 4.94280402173375486122821983506, 5.711620199659219688686482643172, 6.57901774890072509692530803870, 7.418110281714158922866172461632, 8.63756065885991344741182705371, 9.76987068582106773158260793372, 10.05174922357335181063501434576, 11.52333587550906321877702132498, 12.240740257472627596194476776804, 12.805497332746752512241020561247, 13.89023475388347000786845568995, 14.09197164278706285825179071720, 14.863760248135870139774187969401, 16.07662943576029226982501414161, 17.15899755127492462057371257834, 18.06841625931060335587517692385, 18.72967377744792233191571107566, 19.49054635156225394980597859732, 20.39924514840594791801700948362, 20.81694810922795640643519632261, 21.84840993613237088884011150515

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