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Properties

Label 1-2205-2205.592-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.447 + 0.894i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (0.365 + 0.930i)11-s + (0.930 − 0.365i)13-s + (−0.900 + 0.433i)16-s + (0.294 + 0.955i)17-s + (0.5 − 0.866i)19-s + (−0.294 + 0.955i)22-s + (0.680 + 0.733i)23-s + (0.955 + 0.294i)26-s + (−0.955 + 0.294i)29-s + 31-s + (−0.974 − 0.222i)32-s + (−0.365 + 0.930i)34-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (0.365 + 0.930i)11-s + (0.930 − 0.365i)13-s + (−0.900 + 0.433i)16-s + (0.294 + 0.955i)17-s + (0.5 − 0.866i)19-s + (−0.294 + 0.955i)22-s + (0.680 + 0.733i)23-s + (0.955 + 0.294i)26-s + (−0.955 + 0.294i)29-s + 31-s + (−0.974 − 0.222i)32-s + (−0.365 + 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (592, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.363819570 + 3.828503306i\)
\(L(\frac12)\) \(\approx\) \(2.363819570 + 3.828503306i\)
\(L(1)\) \(\approx\) \(1.595339429 + 1.005099942i\)
\(L(1)\) \(\approx\) \(1.595339429 + 1.005099942i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (0.930 - 0.365i)T \)
17 \( 1 + (0.294 + 0.955i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.680 + 0.733i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + T \)
37 \( 1 + (0.680 - 0.733i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
43 \( 1 + (0.563 - 0.826i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.680 + 0.733i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.930 - 0.365i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.930 - 0.365i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−19.34818515330109818166382889794, −18.60582514679547372022906522668, −18.373275937694650273135181730163, −16.993949980357739876245253810531, −16.25444791329872829975186048407, −15.738922576217999898738705785741, −14.65388252124178382535313488890, −14.20017567869144063554371704610, −13.44425867023621423246241152642, −12.90613377758251634804064525015, −11.76952357836179752294880092132, −11.51185650337033338082146930308, −10.699817143146687676500744780587, −9.84060466571452550836830774047, −9.12663619278695889117070451297, −8.26297110486381805779815917071, −7.17501684029763107329185636599, −6.240849008153576101319604020719, −5.765195262819777262702942589853, −4.792334576654661092278961138323, −3.95762055194021254022187570871, −3.23856787553452503118707861543, −2.46472200106921503649099447343, −1.25448056707947481110760518673, −0.69020443580683176560205449967, 0.97897903915797046471875378018, 2.11077642495453578741027091372, 3.09657879640513261570317562323, 3.92856889064210624527584066778, 4.56548919888464896889591712672, 5.630302129883059977109752715711, 6.05243208111942387589492737777, 7.2414062626772798351107073469, 7.466716401627643166924587084, 8.65940147522685713275300909915, 9.17205010087973358841306227939, 10.335408243583012791263035663895, 11.18749695611378213669965237973, 11.87219578041897883587827417106, 12.77812028125093845502163556344, 13.17882992265104473255633591252, 14.04498573466874929276409091073, 14.7844266298596381624635946044, 15.440378195150465258928936066530, 15.916330903533068182605342000327, 16.93433993323316163615984588662, 17.47537948263048493219771555195, 18.05136437320474025005052737426, 19.12654707361949250021259799606, 19.9493905241403772511022657275

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