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Properties

Label 1-847-847.157-r1-0-0
Degree $1$
Conductor $847$
Sign $0.742 + 0.670i$
Analytic cond. $91.0227$
Root an. cond. $91.0227$
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  + (0.953 − 0.299i)2-s + (−0.913 − 0.406i)3-s + (0.820 − 0.572i)4-s + (−0.879 − 0.475i)5-s + (−0.993 − 0.113i)6-s + (0.610 − 0.791i)8-s + (0.669 + 0.743i)9-s + (−0.981 − 0.189i)10-s + (−0.981 + 0.189i)12-s + (0.921 − 0.389i)13-s + (0.610 + 0.791i)15-s + (0.345 − 0.938i)16-s + (−0.964 + 0.263i)17-s + (0.861 + 0.508i)18-s + (0.625 + 0.780i)19-s + (−0.993 + 0.113i)20-s + ⋯
L(s)  = 1  + (0.953 − 0.299i)2-s + (−0.913 − 0.406i)3-s + (0.820 − 0.572i)4-s + (−0.879 − 0.475i)5-s + (−0.993 − 0.113i)6-s + (0.610 − 0.791i)8-s + (0.669 + 0.743i)9-s + (−0.981 − 0.189i)10-s + (−0.981 + 0.189i)12-s + (0.921 − 0.389i)13-s + (0.610 + 0.791i)15-s + (0.345 − 0.938i)16-s + (−0.964 + 0.263i)17-s + (0.861 + 0.508i)18-s + (0.625 + 0.780i)19-s + (−0.993 + 0.113i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.742 + 0.670i$
Analytic conductor: \(91.0227\)
Root analytic conductor: \(91.0227\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (1:\ ),\ 0.742 + 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103386326 + 0.4243479478i\)
\(L(\frac12)\) \(\approx\) \(1.103386326 + 0.4243479478i\)
\(L(1)\) \(\approx\) \(1.061530909 - 0.3690848812i\)
\(L(1)\) \(\approx\) \(1.061530909 - 0.3690848812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.953 - 0.299i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (0.921 - 0.389i)T \)
17 \( 1 + (-0.964 + 0.263i)T \)
19 \( 1 + (0.625 + 0.780i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (-0.483 + 0.875i)T \)
37 \( 1 + (-0.290 + 0.956i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.861 + 0.508i)T \)
53 \( 1 + (0.345 + 0.938i)T \)
59 \( 1 + (0.948 - 0.318i)T \)
61 \( 1 + (0.217 + 0.976i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (0.969 + 0.244i)T \)
79 \( 1 + (-0.432 - 0.901i)T \)
83 \( 1 + (-0.897 + 0.441i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (0.0285 - 0.999i)T \)
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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

−22.09201658633837617670319679966, −21.321444117039549745150901266048, −20.37017485322960934404280608612, −19.692129169587201402634703727804, −18.40000670741555899509191724159, −17.7994026681758478665855370365, −16.6241456607792626595902167837, −16.06541213764748788549632672137, −15.434231978914858857288353190488, −14.810007668729163125055700656278, −13.64791810300113793918748300479, −12.9300564046329338485384896231, −11.79949811319023668676840706262, −11.36467632400109108591408785824, −10.87044477437476403508115603028, −9.57825911864424569244098409310, −8.3401865319730612061427150057, −7.23638681463257693111547214328, −6.67010464401064190044137499760, −5.75852332983894698390630984517, −4.83282507362369774122814498707, −3.95695438568047034186632142601, −3.39351497472388846910036975554, −1.91847661809649907783451555013, −0.23228165660288038755026608708, 0.99878732642117991633685423120, 1.88854069868962059438195113311, 3.412280975495971893452483549852, 4.18991358825621762878564719498, 5.08050943453431041689706614655, 5.879538199551648610153078909932, 6.73896035750136521855629847584, 7.627839014953101995182943699454, 8.59023888309014873117559560656, 10.10397727688835542888180904905, 10.89749682241889559677209078592, 11.57418462109745335479630051335, 12.23223499795353606438520051408, 12.9623191285714592263061204439, 13.59719462727146600662871688634, 14.75753170035906162785360918623, 15.760764290054115343997685611056, 16.12548793906758837932821481652, 16.9952535710565303585806575370, 18.21616953867492394568730462862, 18.827043881868215975229984254367, 19.84540639406029084739206821724, 20.40663283423912836754614587778, 21.2840627046788173508396973201, 22.41498633690630844932524687615

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