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Properties

Label 1-847-847.292-r0-0-0
Degree $1$
Conductor $847$
Sign $0.187 + 0.982i$
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  + (0.991 + 0.132i)2-s + (0.978 + 0.207i)3-s + (0.964 + 0.263i)4-s + (−0.820 + 0.572i)5-s + (0.941 + 0.336i)6-s + (0.921 + 0.389i)8-s + (0.913 + 0.406i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (−0.362 + 0.931i)13-s + (−0.921 + 0.389i)15-s + (0.861 + 0.508i)16-s + (0.272 + 0.962i)17-s + (0.851 + 0.524i)18-s + (−0.830 − 0.556i)19-s + (−0.941 + 0.336i)20-s + ⋯
L(s)  = 1  + (0.991 + 0.132i)2-s + (0.978 + 0.207i)3-s + (0.964 + 0.263i)4-s + (−0.820 + 0.572i)5-s + (0.941 + 0.336i)6-s + (0.921 + 0.389i)8-s + (0.913 + 0.406i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (−0.362 + 0.931i)13-s + (−0.921 + 0.389i)15-s + (0.861 + 0.508i)16-s + (0.272 + 0.962i)17-s + (0.851 + 0.524i)18-s + (−0.830 − 0.556i)19-s + (−0.941 + 0.336i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.635049753 + 2.178643797i\)
\(L(\frac12)\) \(\approx\) \(2.635049753 + 2.178643797i\)
\(L(1)\) \(\approx\) \(2.143588577 + 0.8539677035i\)
\(L(1)\) \(\approx\) \(2.143588577 + 0.8539677035i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.991 + 0.132i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (-0.820 + 0.572i)T \)
13 \( 1 + (-0.362 + 0.931i)T \)
17 \( 1 + (0.272 + 0.962i)T \)
19 \( 1 + (-0.830 - 0.556i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.449 + 0.893i)T \)
37 \( 1 + (-0.935 - 0.353i)T \)
41 \( 1 + (-0.564 - 0.825i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.851 - 0.524i)T \)
53 \( 1 + (0.861 - 0.508i)T \)
59 \( 1 + (0.432 + 0.901i)T \)
61 \( 1 + (0.380 - 0.924i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (0.953 - 0.299i)T \)
79 \( 1 + (0.290 + 0.956i)T \)
83 \( 1 + (0.198 - 0.980i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.0855 + 0.996i)T \)
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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

−21.93868220061391296995669694220, −20.78900229531672563775100267096, −20.56795129369120626434759415777, −19.692272967791948160648239405160, −19.20134916387770419868605111228, −18.16980378345068139377008183597, −16.82580614662654101206026314288, −15.943812575874384112730078698643, −15.3661785594278275203213447018, −14.62633460700063295280617910191, −13.850418209600126836685096674901, −12.980880462762706946605406378159, −12.36627418400291409772639326314, −11.7086868730775013736528366340, −10.48655897306237133418532722117, −9.663215363336286864991506224578, −8.39240393942015626756451513497, −7.79244586208041087953161135077, −7.0028843844485956344290629141, −5.81848671737028517393780568346, −4.71642273861768574916628818468, −4.00410747156064003606314699857, −3.12663552641448482669308167396, −2.25967100816000290801135343286, −0.99100061446377897539950577647, 1.85719589985352742634566123307, 2.61350823047946623863190922062, 3.79270535893117738183574694839, 4.04397497157978128376305631317, 5.187871618660682277238163300357, 6.61820860363809898892196393954, 7.11115579353479197369663306303, 8.10101475991250327347515225116, 8.80626487542352753596345546925, 10.277945204951778424294345800374, 10.785222501155677517554877049217, 12.01530581783799370081442985527, 12.49505823329821338540592422548, 13.69513537071911720953622803046, 14.254034085252301168284754601128, 14.92509730513391057832899655639, 15.61554582030744779483482790903, 16.23088094820872180548191827258, 17.299761882177859230020619801880, 18.65677265358048268452340360951, 19.536784435233416749676529019040, 19.73426141943108408014108362890, 20.83964145873901949231800763769, 21.6617851745719418824923803753, 22.02288678970354665161937513186

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