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Properties

Label 1-2205-2205.1082-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.894 - 0.447i$
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 + (−0.365 − 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

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

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1952072507 - 0.8252882824i\)
\(L(\frac12)\) \(\approx\) \(0.1952072507 - 0.8252882824i\)
\(L(1)\) \(\approx\) \(0.6393541155 - 0.3472695592i\)
\(L(1)\) \(\approx\) \(0.6393541155 - 0.3472695592i\)

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

−19.93304186937819477015876734425, −19.17918828934438447035804724091, −18.45344547793529943291754835718, −17.85749745955159536110299411489, −17.25296381678371911264764922035, −16.45035238374096414333663826883, −15.6538720020018454012228008695, −15.33016137124874813925874364818, −14.289431683083916496030341131700, −13.77967904593571558611160867839, −12.771706848961230672913170018165, −11.89010388463180559964586084620, −11.06931171060741447187051543647, −10.2490854012890793184204065149, −9.753955020200236122140612448193, −8.86739048657864454460182865769, −8.0377763615636884516052086119, −7.62093105093647502122314128459, −6.39337533114382469314811772003, −6.170778669240283137108152832022, −5.00944402882632605353715271954, −4.27300702016554479750060667741, −3.09317146128188322917036396116, −1.83719633827770655474945467894, −1.308138333385050686013016556847, 0.4117463756175622386612517358, 1.20067214400979473373504944462, 2.53336776876182051976147037218, 2.9868893936705869958961741678, 3.980497039271431622265165705369, 4.90159241056230943800288529025, 6.03674633357725598844437065227, 6.79728986692050411742348214883, 7.742888349532263119688280476053, 8.46714164452445809637150585906, 8.96042986605372772586589473360, 9.967322391505535606597304854580, 10.54140867891463327378854284831, 11.418982349192437934635192165813, 11.75655083649009914720332920288, 12.86491162657813073844405614334, 13.49763173648642600069164465471, 14.067805319165379295167429009829, 15.46088508056978583071738004129, 15.9940830951502239576093524396, 16.49656876021579689097209118843, 17.54273791595229386938255947271, 18.09918642400218860309375909277, 18.617843903767157308793287964877, 19.42430694053887651683664468481

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