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Properties

Label 1-2205-2205.853-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.686 - 0.726i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.433 − 0.900i)8-s + (0.365 − 0.930i)11-s + (0.149 + 0.988i)13-s + (0.0747 + 0.997i)16-s + (0.974 − 0.222i)17-s + 19-s + (−0.680 + 0.733i)22-s + (0.680 − 0.733i)23-s + (0.222 − 0.974i)26-s + (0.733 − 0.680i)29-s + (0.5 − 0.866i)31-s + (0.294 − 0.955i)32-s + (−0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.433 − 0.900i)8-s + (0.365 − 0.930i)11-s + (0.149 + 0.988i)13-s + (0.0747 + 0.997i)16-s + (0.974 − 0.222i)17-s + 19-s + (−0.680 + 0.733i)22-s + (0.680 − 0.733i)23-s + (0.222 − 0.974i)26-s + (0.733 − 0.680i)29-s + (0.5 − 0.866i)31-s + (0.294 − 0.955i)32-s + (−0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.686 - 0.726i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.686 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.105077048 - 0.4760978174i\)
\(L(\frac12)\) \(\approx\) \(1.105077048 - 0.4760978174i\)
\(L(1)\) \(\approx\) \(0.8077265966 - 0.1663172836i\)
\(L(1)\) \(\approx\) \(0.8077265966 - 0.1663172836i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.930 - 0.365i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.149 + 0.988i)T \)
17 \( 1 + (0.974 - 0.222i)T \)
19 \( 1 + T \)
23 \( 1 + (0.680 - 0.733i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 + (-0.0747 + 0.997i)T \)
43 \( 1 + (-0.997 + 0.0747i)T \)
47 \( 1 + (0.930 + 0.365i)T \)
53 \( 1 + (-0.974 - 0.222i)T \)
59 \( 1 + (0.826 - 0.563i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.781 - 0.623i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.149 + 0.988i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−19.74131390112351380694944313235, −19.05354046028189626269607739430, −18.282437601770916147720491460875, −17.50199693069965495322579153533, −17.28094444560005909671857097961, −16.14767882476500232621538664470, −15.68477543281942673633413593039, −14.897009791911922600026096895896, −14.29354494153823916044054824839, −13.36076762521841951971667191007, −12.18940716112613521971065789037, −11.90472297844077284496338711828, −10.64023350596724498118170044184, −10.269183325635087757368745041823, −9.45551934079197123218783545321, −8.74043028122117976813213735013, −7.88169455435110563408414234, −7.24838502341039466848363431520, −6.594388460215306666250496160241, −5.43843290174891958654656949655, −5.10553126098343765403679774781, −3.590699846102386374117798976727, −2.824226620040879613560942517382, −1.62519346113676518698072413932, −0.948163368033607606253989097785, 0.733543468770659780397149157, 1.48083738783550964019504995724, 2.64341648413324352276954556255, 3.32676842286535643370984765039, 4.21588297144350955459903476128, 5.38960718880188692022828864372, 6.41266974759815240852771393345, 6.94293425404619222128629836370, 8.01017271621058531221510368686, 8.48304646937285351165896990664, 9.43271001663298285857174606650, 9.87838407116291912488987474752, 10.85140802007968144416948694467, 11.61737650965370720730969280218, 11.93603739100632199087529137302, 12.9888417319981749750803001468, 13.85007447998688449515089879136, 14.51502379921542335613622154449, 15.6262540875621591974542381456, 16.23768664279455147458802099909, 16.831638369864909392916960399770, 17.42520501819471521160565715054, 18.45247679985924585233270991952, 18.9283144985970391203775972865, 19.34259523808417858586080454145

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