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Properties

Label 1-847-847.206-r0-0-0
Degree $1$
Conductor $847$
Sign $0.670 - 0.742i$
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  + (−0.953 + 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

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

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.670 - 0.742i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.670 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3728878793 - 0.1657390197i\)
\(L(\frac12)\) \(\approx\) \(0.3728878793 - 0.1657390197i\)
\(L(1)\) \(\approx\) \(0.4331711755 - 0.04756144488i\)
\(L(1)\) \(\approx\) \(0.4331711755 - 0.04756144488i\)

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

−22.18557052648746429925957310598, −21.340745722533955415977226620046, −20.64226027455951430090235964786, −19.503429655566697046517756173447, −19.14000444795081296436817396948, −18.08951451097424479578207414184, −17.55825636423566978818077486, −16.5800935587697620659322897355, −16.11672694532007537686694221916, −15.190380462288469445994914751325, −14.55410493667094007936400928269, −12.76693087517835699545873749216, −12.06548713813529553115665048895, −11.62784373162198983056635266816, −10.45169068048963644592596225083, −10.28293161171167630539385853084, −9.193025464820681413828916744, −7.9609561623435690046855929455, −7.435837757794128312555098100245, −6.433687904757867696175655570968, −5.511631010688683222055264075973, −4.120878772212371536172233015351, −3.449560399882146645769038203414, −2.15389575041301048337197778607, −0.68935788186567631548310971158, 0.48614824376777795765237709702, 1.54223909730120941017029025596, 2.81092285507005442995431238155, 4.486029622037796937736595158037, 5.17970014938787789864598812898, 6.32516312939505374002488174895, 7.08824371359261544980525997926, 7.80992277350470169802070971015, 8.60013215833640877475522589748, 9.703486766712296378468162990815, 10.5247604841063310234811266949, 11.39762695567209632663192614097, 12.09294720941241244373265559962, 12.62967999489240405643851853850, 14.081216884572102653090870421868, 15.036222400695287542823485538406, 16.00284979165997147833299078408, 16.450931860653594204373254883080, 17.1950532203413212148042829629, 17.91358588538005917026412759128, 18.84173268141672555028108082855, 19.381367643464045091703703913274, 20.06995071468807725632710805541, 21.13176146163798696795804250172, 22.04913980716470799424400267840

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