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Properties

Label 1-2205-2205.1163-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.972 - 0.234i$
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  + (−0.563 + 0.826i)2-s + (−0.365 − 0.930i)4-s + (0.974 + 0.222i)8-s + (−0.826 − 0.563i)11-s + (0.997 + 0.0747i)13-s + (−0.733 + 0.680i)16-s + (−0.781 + 0.623i)17-s − 19-s + (0.930 − 0.365i)22-s + (0.930 − 0.365i)23-s + (−0.623 + 0.781i)26-s + (0.365 − 0.930i)29-s + (−0.5 + 0.866i)31-s + (−0.149 − 0.988i)32-s + (−0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (−0.563 + 0.826i)2-s + (−0.365 − 0.930i)4-s + (0.974 + 0.222i)8-s + (−0.826 − 0.563i)11-s + (0.997 + 0.0747i)13-s + (−0.733 + 0.680i)16-s + (−0.781 + 0.623i)17-s − 19-s + (0.930 − 0.365i)22-s + (0.930 − 0.365i)23-s + (−0.623 + 0.781i)26-s + (0.365 − 0.930i)29-s + (−0.5 + 0.866i)31-s + (−0.149 − 0.988i)32-s + (−0.0747 − 0.997i)34-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8812881634 - 0.1047300972i\)
\(L(\frac12)\) \(\approx\) \(0.8812881634 - 0.1047300972i\)
\(L(1)\) \(\approx\) \(0.7211311181 + 0.1598906562i\)
\(L(1)\) \(\approx\) \(0.7211311181 + 0.1598906562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.563 + 0.826i)T \)
11 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (0.997 + 0.0747i)T \)
17 \( 1 + (-0.781 + 0.623i)T \)
19 \( 1 - T \)
23 \( 1 + (0.930 - 0.365i)T \)
29 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.781 - 0.623i)T \)
41 \( 1 + (0.733 + 0.680i)T \)
43 \( 1 + (-0.680 - 0.733i)T \)
47 \( 1 + (-0.563 + 0.826i)T \)
53 \( 1 + (-0.781 - 0.623i)T \)
59 \( 1 + (0.955 + 0.294i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.433 - 0.900i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.997 - 0.0747i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
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

−19.78233542831598223991181492856, −19.02376275863096407284890155699, −18.253112236373807420958509343597, −17.9191179593748039163702502868, −17.02133058426544705238258536362, −16.28845383082188447463419846715, −15.5506512598296399654699878743, −14.74293750753535554427106890700, −13.56467146441465621993473436848, −13.10313727801439431826139154411, −12.532738406897434181615273618019, −11.43310668660792223573577452809, −11.001107108862677025006030659231, −10.29253157367413807918586202032, −9.440272513165634078197941699493, −8.76347839052261430924174133963, −8.060142224510641204114971444850, −7.22910076381864906906456732710, −6.42609202684939633049812970847, −5.17277244350915770903689676671, −4.45612508515631227748399642006, −3.54984619529762194637045485057, −2.66173086142805112217624783331, −1.93176853400665333234214493248, −0.86189690458273238657918667581, 0.46631906584526683170657459515, 1.60134919028007764319637463994, 2.60897973552601662175591990052, 3.857356012929002849186152759338, 4.6855299177954430163222348516, 5.582408994687705637551509369544, 6.32581902772656200915254978752, 6.87458337226662589585126692902, 8.05814501455824910264747708967, 8.4044081382135305737544517328, 9.14000286179428269185372047347, 10.0792124308988364042026586664, 10.98295622411052279777862616290, 11.119893392786912179399263398757, 12.78619864308455250682410577174, 13.18363772164519457251336478242, 14.028736099035564186958435435138, 14.83779320976074249443036181915, 15.48843452943938003094240915974, 16.13658284473391522729100714239, 16.73078207975463812924173660487, 17.6453478734397015728734339968, 18.07905399747842339815809416773, 19.03276479947832277942096259764, 19.30025706905070912304378462912

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