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Properties

Label 1-2205-2205.403-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.234 - 0.972i$
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  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.974 − 0.222i)8-s + (0.0747 − 0.997i)11-s + (0.997 + 0.0747i)13-s + (−0.222 − 0.974i)16-s + (−0.930 − 0.365i)17-s + (0.5 − 0.866i)19-s + (0.930 − 0.365i)22-s + (0.149 − 0.988i)23-s + (0.365 + 0.930i)26-s + (−0.365 + 0.930i)29-s + 31-s + (0.781 − 0.623i)32-s + (−0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.974 − 0.222i)8-s + (0.0747 − 0.997i)11-s + (0.997 + 0.0747i)13-s + (−0.222 − 0.974i)16-s + (−0.930 − 0.365i)17-s + (0.5 − 0.866i)19-s + (0.930 − 0.365i)22-s + (0.149 − 0.988i)23-s + (0.365 + 0.930i)26-s + (−0.365 + 0.930i)29-s + 31-s + (0.781 − 0.623i)32-s + (−0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.234 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3544444666 - 0.4500484017i\)
\(L(\frac12)\) \(\approx\) \(0.3544444666 - 0.4500484017i\)
\(L(1)\) \(\approx\) \(1.021548043 + 0.3843065867i\)
\(L(1)\) \(\approx\) \(1.021548043 + 0.3843065867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.997 + 0.0747i)T \)
17 \( 1 + (-0.930 - 0.365i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.149 - 0.988i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 + T \)
37 \( 1 + (0.149 + 0.988i)T \)
41 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (-0.680 - 0.733i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (0.149 - 0.988i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.997 + 0.0747i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.997 + 0.0747i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
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

−20.03935321155380006363286763534, −19.09808676639383363603521901873, −18.461196881819888978941625320900, −17.74400890926192396170369844569, −17.153446673221375239561320452974, −15.87204472206622650087567336933, −15.346267091741225798091395861685, −14.60063080981910494645269277506, −13.72877949352121745591920847405, −13.186275656336609022130808534776, −12.49957528124106906409753819050, −11.60283841680354533936058880751, −11.19778012465352529388125688495, −10.12874242464561168286573983816, −9.74059436550478349075999652797, −8.79235157995864191015088642838, −8.030434353401607578372907670340, −6.90706649359017635924144194673, −6.04065327752425865518709062743, −5.327278635958443580428243286633, −4.30640479488614894724082846106, −3.820050345993039812119978689002, −2.81972374965183206090569446586, −1.85634274051498104559062222164, −1.218395520952162884444628060157, 0.08975341964190851325191132004, 1.08205631501091246945723648795, 2.63339169746469900613363294770, 3.344016191298891308559364673365, 4.24680620560792674991594347930, 5.02437385699258069928053224754, 5.83807147820049076841318945913, 6.63404635139878290367983616970, 7.11049840060461268391068554008, 8.4811394667752836758064829627, 8.52426569064143071774397690993, 9.466472050269180093061146629996, 10.63765939594487471433380828772, 11.38805419100987488336179951193, 12.104717895092382045726999246323, 13.20887243213371709974818529595, 13.53298711153255431025276328152, 14.192696722223417228280440982901, 15.144584151377444279563900720396, 15.744843969032454612706699529598, 16.34443311844499158557420997393, 17.00075722920387290984740125014, 17.880238090804933309984877171115, 18.43174798754152609712851798582, 19.14080490331739183158081400842

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