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Properties

Label 1-847-847.299-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.729 - 0.684i$
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  + (0.179 + 0.983i)2-s + (−0.669 − 0.743i)3-s + (−0.935 + 0.353i)4-s + (0.290 + 0.956i)5-s + (0.610 − 0.791i)6-s + (−0.516 − 0.856i)8-s + (−0.104 + 0.994i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (−0.998 − 0.0570i)13-s + (0.516 − 0.856i)15-s + (0.749 − 0.662i)16-s + (0.999 + 0.0380i)17-s + (−0.997 + 0.0760i)18-s + (0.345 + 0.938i)19-s + (−0.610 − 0.791i)20-s + ⋯
L(s)  = 1  + (0.179 + 0.983i)2-s + (−0.669 − 0.743i)3-s + (−0.935 + 0.353i)4-s + (0.290 + 0.956i)5-s + (0.610 − 0.791i)6-s + (−0.516 − 0.856i)8-s + (−0.104 + 0.994i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (−0.998 − 0.0570i)13-s + (0.516 − 0.856i)15-s + (0.749 − 0.662i)16-s + (0.999 + 0.0380i)17-s + (−0.997 + 0.0760i)18-s + (0.345 + 0.938i)19-s + (−0.610 − 0.791i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.729 - 0.684i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.729 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1266930497 + 0.3202555484i\)
\(L(\frac12)\) \(\approx\) \(-0.1266930497 + 0.3202555484i\)
\(L(1)\) \(\approx\) \(0.5795489174 + 0.3772706884i\)
\(L(1)\) \(\approx\) \(0.5795489174 + 0.3772706884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.179 + 0.983i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (0.290 + 0.956i)T \)
13 \( 1 + (-0.998 - 0.0570i)T \)
17 \( 1 + (0.999 + 0.0380i)T \)
19 \( 1 + (0.345 + 0.938i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (-0.988 - 0.151i)T \)
37 \( 1 + (0.964 - 0.263i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.997 - 0.0760i)T \)
53 \( 1 + (0.749 + 0.662i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.761 - 0.647i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (-0.595 + 0.803i)T \)
79 \( 1 + (-0.820 + 0.572i)T \)
83 \( 1 + (-0.870 - 0.491i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.974 + 0.226i)T \)
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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

−21.56235867706013410392097199308, −20.93733738009963609039291240189, −20.16210079359420156019041554304, −19.570612732358606752707343735419, −18.335256842483531744907121764098, −17.629029807056809560099062661616, −16.87815362462452325302727226872, −16.211857219951689225219540388347, −15.06104360001269087436753777557, −14.298538069430894165468011053340, −13.255059180641875415862244856720, −12.42397429493459649075512026168, −11.87447896303278906928195761853, −11.06062534838609475538211306186, −9.97495281664412493213251227064, −9.59436069945258072254206707547, −8.80568513252759122584930451828, −7.58556942520500323584082539256, −5.96568957172410666832660348585, −5.320818456341056112457544479551, −4.59329698140245501299507575449, −3.80470576924280853563815779610, −2.60514315283449932124876924434, −1.35950144668190550196729979025, −0.166128604029968040879536302353, 1.61569821497007750678169291038, 2.90151021736680337574818998017, 4.045235991441284138036312965386, 5.40562969917957210911376234344, 5.81181388430537566885303983897, 6.75057001796751809114944142341, 7.55386144979585392613303839284, 7.950266932151385858362964025, 9.53033397576686644178511873251, 10.17017058128597494709700043721, 11.301236077821225276745788625205, 12.274415668426249942793241514701, 12.87793282004366926749001786904, 14.04078110923954283882403349758, 14.36980945844631978853747714430, 15.28632197257590391803948634917, 16.499930404353310504966969412779, 16.86501173666633595121044224090, 17.86768109017097252254103050503, 18.41080051672426065045635059145, 18.94212132195624791176084653202, 20.041072784803496367716258344856, 21.6116674851948195991067961739, 21.96147654255481947676541126383, 22.8725679907641771879893868038

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