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Properties

Label 1-847-847.439-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.647 + 0.762i$
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  + (−0.928 + 0.371i)2-s + (0.5 − 0.866i)3-s + (0.723 − 0.690i)4-s + (−0.580 + 0.814i)5-s + (−0.142 + 0.989i)6-s + (−0.415 + 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.235 − 0.971i)10-s + (−0.235 − 0.971i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)15-s + (0.0475 − 0.998i)16-s + (−0.327 − 0.945i)17-s + (0.786 + 0.618i)18-s + (−0.327 + 0.945i)19-s + (0.142 + 0.989i)20-s + ⋯
L(s)  = 1  + (−0.928 + 0.371i)2-s + (0.5 − 0.866i)3-s + (0.723 − 0.690i)4-s + (−0.580 + 0.814i)5-s + (−0.142 + 0.989i)6-s + (−0.415 + 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.235 − 0.971i)10-s + (−0.235 − 0.971i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)15-s + (0.0475 − 0.998i)16-s + (−0.327 − 0.945i)17-s + (0.786 + 0.618i)18-s + (−0.327 + 0.945i)19-s + (0.142 + 0.989i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.647 + 0.762i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.647 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1078188370 + 0.2330263997i\)
\(L(\frac12)\) \(\approx\) \(0.1078188370 + 0.2330263997i\)
\(L(1)\) \(\approx\) \(0.5770590097 + 0.01113182222i\)
\(L(1)\) \(\approx\) \(0.5770590097 + 0.01113182222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.928 + 0.371i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (-0.327 + 0.945i)T \)
23 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (0.723 + 0.690i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.786 - 0.618i)T \)
53 \( 1 + (0.0475 + 0.998i)T \)
59 \( 1 + (-0.928 - 0.371i)T \)
61 \( 1 + (-0.786 + 0.618i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
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

−21.63402628512394576044475157622, −20.84413661996493174298736021556, −20.11942260145236491547539496109, −19.44153071458233002595485598239, −19.12527083577754936327263016249, −17.52414422125967627815212717807, −17.09057541271000548382598038471, −16.29846991183088361490040864445, −15.43793930942507598519964287097, −15.05106852701194676702560690046, −13.61729137079524981513858670156, −12.734611455405207590298742409069, −11.79250205502490622493326037601, −11.08744028846659031055385007622, −10.1641345898354708579655702005, −9.31859291182684599568219048040, −8.81001450445389592410092090906, −7.93780240641522586729219143311, −7.23419036815125010949135403785, −5.753122642640357747220414219997, −4.50422234103927148569846586125, −3.90225320820250883127291839562, −2.73912364452291746019122152716, −1.78921131731641096846332621, −0.1478142974155387147168078399, 1.28226332067308417038924850594, 2.54458336498652798569037580229, 3.055863355475874145975508001974, 4.70439408032115353175941596803, 6.114820541635589968373687970068, 6.80307129586462702079765375704, 7.49771537344972611990865648838, 8.13756239600870207520887919876, 9.00431678696464853045354106725, 10.02431703623266045173897781036, 10.806481487030205140284420375569, 11.85716654811056410580317145748, 12.382679168930848642446097776783, 13.77582598502410813637806042672, 14.62971520391875712700488734438, 14.95593419367336148764629310141, 16.05771119718610625223904091649, 16.89661060241913593311649705385, 17.96291701534502687736397111332, 18.393658018272314127585834862200, 19.02280561997161481881872889528, 19.99639460403083234168702340707, 20.13510064344482895343323355832, 21.49839629569770508008903455387, 22.72952358236653308426058947526

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