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Properties

Label 1-2205-2205.1438-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.285 + 0.958i$
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  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.433 − 0.900i)8-s + (−0.988 + 0.149i)11-s + (0.149 + 0.988i)13-s + (−0.900 − 0.433i)16-s + (−0.680 − 0.733i)17-s + (−0.5 + 0.866i)19-s + (−0.680 + 0.733i)22-s + (0.294 + 0.955i)23-s + (0.733 + 0.680i)26-s + (0.733 − 0.680i)29-s − 31-s + (−0.974 + 0.222i)32-s + (−0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.433 − 0.900i)8-s + (−0.988 + 0.149i)11-s + (0.149 + 0.988i)13-s + (−0.900 − 0.433i)16-s + (−0.680 − 0.733i)17-s + (−0.5 + 0.866i)19-s + (−0.680 + 0.733i)22-s + (0.294 + 0.955i)23-s + (0.733 + 0.680i)26-s + (0.733 − 0.680i)29-s − 31-s + (−0.974 + 0.222i)32-s + (−0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.285 + 0.958i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1438, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.285 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6866561540 + 0.5116738752i\)
\(L(\frac12)\) \(\approx\) \(0.6866561540 + 0.5116738752i\)
\(L(1)\) \(\approx\) \(1.147214947 - 0.3366354518i\)
\(L(1)\) \(\approx\) \(1.147214947 - 0.3366354518i\)

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 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.149 + 0.988i)T \)
17 \( 1 + (-0.680 - 0.733i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.294 + 0.955i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (0.294 - 0.955i)T \)
41 \( 1 + (-0.0747 + 0.997i)T \)
43 \( 1 + (-0.997 + 0.0747i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (0.294 + 0.955i)T \)
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 \( 1 + (-0.149 + 0.988i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.149 + 0.988i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
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

−19.88774717838865657870692165708, −18.68983906200171930689779951954, −17.961290009326947492285970398448, −17.37574029878025820087083028140, −16.544806762466250785612552398972, −15.84033874125448295345124889354, −15.17738351280171892048357660392, −14.727191262267555432863124056298, −13.673254817872223642779189819475, −12.98269976634372854189191367826, −12.73685517061567332992843237746, −11.64245785475538514260889070102, −10.789316080645891502492014756524, −10.27324016599409801284982542924, −8.77756275973781808126984614049, −8.44037219816931844571208376370, −7.55173374935991551542192283651, −6.73345558803488566738302750352, −6.05513602037769964366250145939, −5.112155432764567476210610955043, −4.6587453436074171631452675853, −3.514278564955638285281630924624, −2.85050492009667874980669818137, −1.95288405586831383119517183438, −0.19628071063387000122146120937, 1.3410311987925311135427606262, 2.17978833389926953662184507327, 2.94054329124088096497973026822, 3.93404607564357148629341219404, 4.62280392115128920637509510070, 5.40163093178807296878659626978, 6.219179049410603750972304095409, 7.01481525074640713574899965285, 7.90513328756480227057591414746, 9.03118462860917888034144862871, 9.69301082511716330525854896006, 10.47885770257855937610231236788, 11.2629814016099314938039759359, 11.76482380048253968546211749944, 12.72582233193358323858253028307, 13.25550189989317892130111743096, 13.98283937520001331816352598087, 14.645503773566832231197170928129, 15.50649655941708221953466867920, 16.037422492700765664318550867517, 16.8815345329848840461054917773, 18.0779776878621620919630635978, 18.47609686735492752891389541062, 19.32062542662093218586097546100, 19.954735530446642036785393139934

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