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Properties

Label 1-847-847.629-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.201 - 0.979i$
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  + (0.466 − 0.884i)2-s + (−0.309 + 0.951i)3-s + (−0.564 − 0.825i)4-s + (−0.198 − 0.980i)5-s + (0.696 + 0.717i)6-s + (−0.993 + 0.113i)8-s + (−0.809 − 0.587i)9-s + (−0.959 − 0.281i)10-s + (0.959 − 0.281i)12-s + (0.941 + 0.336i)13-s + (0.993 + 0.113i)15-s + (−0.362 + 0.931i)16-s + (0.974 + 0.226i)17-s + (−0.897 + 0.441i)18-s + (0.516 + 0.856i)19-s + (−0.696 + 0.717i)20-s + ⋯
L(s)  = 1  + (0.466 − 0.884i)2-s + (−0.309 + 0.951i)3-s + (−0.564 − 0.825i)4-s + (−0.198 − 0.980i)5-s + (0.696 + 0.717i)6-s + (−0.993 + 0.113i)8-s + (−0.809 − 0.587i)9-s + (−0.959 − 0.281i)10-s + (0.959 − 0.281i)12-s + (0.941 + 0.336i)13-s + (0.993 + 0.113i)15-s + (−0.362 + 0.931i)16-s + (0.974 + 0.226i)17-s + (−0.897 + 0.441i)18-s + (0.516 + 0.856i)19-s + (−0.696 + 0.717i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.201 - 0.979i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8876781742 - 1.088465083i\)
\(L(\frac12)\) \(\approx\) \(0.8876781742 - 1.088465083i\)
\(L(1)\) \(\approx\) \(0.9949064722 - 0.5067025904i\)
\(L(1)\) \(\approx\) \(0.9949064722 - 0.5067025904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.466 - 0.884i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.198 - 0.980i)T \)
13 \( 1 + (0.941 + 0.336i)T \)
17 \( 1 + (0.974 + 0.226i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (-0.610 - 0.791i)T \)
37 \( 1 + (-0.0285 - 0.999i)T \)
41 \( 1 + (-0.985 - 0.170i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.897 - 0.441i)T \)
53 \( 1 + (-0.362 - 0.931i)T \)
59 \( 1 + (0.985 - 0.170i)T \)
61 \( 1 + (-0.466 - 0.884i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (0.774 + 0.633i)T \)
79 \( 1 + (0.870 - 0.491i)T \)
83 \( 1 + (-0.998 + 0.0570i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.198 + 0.980i)T \)
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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

−22.61321357029040799542021730477, −21.88448478952461123634803093502, −20.962702907948901035900546309704, −19.73893897176989004853440885637, −18.85498929142775639000916226023, −18.16800054542307402372230549243, −17.68339977943546207456824412866, −16.72341573820029367423123158375, −15.84176273694655784594936514390, −15.05449389974778252037754303857, −14.161271320678326055932671805, −13.57670733333999915322878248390, −12.83075482903470671422064271083, −11.779438244011117551915163184206, −11.252016526113570959874560478457, −10.03599157059721010599345598980, −8.72016268005483777741060397267, −7.912543087130799893974250692631, −7.16431848008505974054312456356, −6.541294171790754786610404024914, −5.73552856944427565930901890899, −4.84458767907547722283206816958, −3.33687849318026006108005519592, −2.87074309046720608349707006680, −1.14988941905933100342554779473, 0.67966814160027836238691425616, 1.79107984661021203050209379304, 3.405151786578308619869202325736, 3.81203952647562626115965212801, 4.9176884614453260771338601105, 5.42607150236073093266050303151, 6.38461351374075956059501118698, 8.17697634996180942219557350818, 8.91858178978711326507552304980, 9.68494176633832791363938613489, 10.46992772304103749459428470017, 11.324463257937921383347924942927, 12.05245387691542913846425958256, 12.73121341908050926914059586330, 13.75701259520439353847062395604, 14.55111195985018133284855165787, 15.4263326245475326837716391039, 16.30926847818778899341247806577, 16.86735529513651462996059886089, 17.99989920074875420147958146613, 18.86180118170564558724628563712, 19.81477625088605297169997378124, 20.63747417528147701359005896928, 20.96223307250466120584412405668, 21.651100459486630883841369800956

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