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Properties

Label 1-847-847.271-r0-0-0
Degree $1$
Conductor $847$
Sign $0.425 - 0.905i$
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  + (0.432 − 0.901i)2-s + (−0.913 + 0.406i)3-s + (−0.625 − 0.780i)4-s + (−0.797 − 0.603i)5-s + (−0.0285 + 0.999i)6-s + (−0.974 + 0.226i)8-s + (0.669 − 0.743i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (0.774 + 0.633i)13-s + (0.974 + 0.226i)15-s + (−0.217 + 0.976i)16-s + (−0.830 + 0.556i)17-s + (−0.380 − 0.924i)18-s + (0.999 − 0.0380i)19-s + (0.0285 + 0.999i)20-s + ⋯
L(s)  = 1  + (0.432 − 0.901i)2-s + (−0.913 + 0.406i)3-s + (−0.625 − 0.780i)4-s + (−0.797 − 0.603i)5-s + (−0.0285 + 0.999i)6-s + (−0.974 + 0.226i)8-s + (0.669 − 0.743i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (0.774 + 0.633i)13-s + (0.974 + 0.226i)15-s + (−0.217 + 0.976i)16-s + (−0.830 + 0.556i)17-s + (−0.380 − 0.924i)18-s + (0.999 − 0.0380i)19-s + (0.0285 + 0.999i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.425 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7796254269 - 0.4950756681i\)
\(L(\frac12)\) \(\approx\) \(0.7796254269 - 0.4950756681i\)
\(L(1)\) \(\approx\) \(0.7312004434 - 0.3533108557i\)
\(L(1)\) \(\approx\) \(0.7312004434 - 0.3533108557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.432 - 0.901i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (-0.797 - 0.603i)T \)
13 \( 1 + (0.774 + 0.633i)T \)
17 \( 1 + (-0.830 + 0.556i)T \)
19 \( 1 + (0.999 - 0.0380i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (-0.696 - 0.717i)T \)
31 \( 1 + (0.710 + 0.703i)T \)
37 \( 1 + (0.548 + 0.836i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.380 + 0.924i)T \)
53 \( 1 + (-0.217 - 0.976i)T \)
59 \( 1 + (0.179 - 0.983i)T \)
61 \( 1 + (0.997 + 0.0760i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.985 - 0.170i)T \)
73 \( 1 + (-0.948 - 0.318i)T \)
79 \( 1 + (0.999 + 0.0190i)T \)
83 \( 1 + (0.993 - 0.113i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (0.921 + 0.389i)T \)
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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

−22.420108230486045194460149795345, −22.065035178587988717927542503036, −20.801604990420301508369800667862, −19.79240207836589763825648981638, −18.64719488257087369404186327003, −18.086384380729249222221578581060, −17.59562565837899720691001890597, −16.25415280028287137681050094072, −16.03995842332850577112025101309, −15.2018419412301539018999088445, −14.18990966458439502702733668838, −13.37102500680292289849374145395, −12.58928366034162231346152705084, −11.67836150217007936837178763584, −11.133282216439362529152513508640, −10.0084229227745388352178849968, −8.72943809599592383179035849701, −7.66718171222913747002091693527, −7.29345210801273763267193807921, −6.25399917203022067147859819482, −5.65946988678469920352150964289, −4.54627713649295550655544749756, −3.76389951792103794315320739837, −2.58013087765948057900407932865, −0.68297388875454963799559510252, 0.739815722709720968345612689403, 1.781475732575982629799671172472, 3.40069544275296173359482655074, 4.15090914615129656130183137176, 4.750194479424466501714858162584, 5.76239584629920921303988202492, 6.56836607255868370562278852941, 7.9984183868446334270965374941, 9.05034201801790971738923501972, 9.76238505169755324936107895012, 10.81492269829214095001158659256, 11.471080690252391235878341059408, 11.95906212900286000277097106487, 12.83655848020506081347753548282, 13.59925859568623703083792161751, 14.73064013142388510726034353239, 15.74129928495057282275924894719, 16.09251489270657546172999961541, 17.29549902324376934114907990927, 18.03014532880600742253077769586, 18.91759373126532874423508705380, 19.69314906482798161697326037198, 20.61422255334356924171719221976, 21.04924067239518364365303015409, 22.12118651947877277343600249574

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