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Properties

Label 1-2205-2205.1633-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.816 + 0.576i$
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  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (−0.733 + 0.680i)11-s + (−0.680 − 0.733i)13-s + (0.623 − 0.781i)16-s + (−0.563 − 0.826i)17-s + (0.5 + 0.866i)19-s + (0.563 − 0.826i)22-s + (0.997 + 0.0747i)23-s + (0.826 + 0.563i)26-s + (−0.826 + 0.563i)29-s + 31-s + (−0.433 + 0.900i)32-s + (0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (−0.733 + 0.680i)11-s + (−0.680 − 0.733i)13-s + (0.623 − 0.781i)16-s + (−0.563 − 0.826i)17-s + (0.5 + 0.866i)19-s + (0.563 − 0.826i)22-s + (0.997 + 0.0747i)23-s + (0.826 + 0.563i)26-s + (−0.826 + 0.563i)29-s + 31-s + (−0.433 + 0.900i)32-s + (0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.816 + 0.576i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.816 + 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8444035691 + 0.2680333591i\)
\(L(\frac12)\) \(\approx\) \(0.8444035691 + 0.2680333591i\)
\(L(1)\) \(\approx\) \(0.6252573589 + 0.05685218930i\)
\(L(1)\) \(\approx\) \(0.6252573589 + 0.05685218930i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.680 - 0.733i)T \)
17 \( 1 + (-0.563 - 0.826i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.997 + 0.0747i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + T \)
37 \( 1 + (0.997 - 0.0747i)T \)
41 \( 1 + (0.365 - 0.930i)T \)
43 \( 1 + (-0.930 + 0.365i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (0.997 + 0.0747i)T \)
59 \( 1 + (-0.623 + 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.680 - 0.733i)T \)
79 \( 1 - T \)
83 \( 1 + (0.680 - 0.733i)T \)
89 \( 1 + (-0.955 + 0.294i)T \)
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

−19.497214201555434802629572459255, −18.71438289965125633263093805816, −18.18372347178292111490289634499, −17.24187833179333321945689331100, −16.8303492172465151897605045244, −16.01546430002587400226928506174, −15.29380911228473939903205322143, −14.6655682615340222730403880671, −13.3675373843639445399197046615, −13.01500471765676340926971614289, −11.842792423616215862178271628548, −11.34915171699045617700967884069, −10.65276171789377288852569191930, −9.82023654621704141335991599585, −9.16024335368806505643598160475, −8.40253158194452396564354999465, −7.70982031732976093031984647849, −6.87435427371644613808001784929, −6.20428775930663373655787936707, −5.143236581697054238185059408, −4.15730491609052821133504982703, −3.0148673881388764683949425152, −2.4456196235104063680799667471, −1.39838118577061360916646519581, −0.385614000625872142553909419004, 0.50443402539228807702347540234, 1.57146665834074660195594217340, 2.544694500078276153969465829364, 3.18348727683566885685972711825, 4.699984006695584206325097787192, 5.33720229339811730345436554160, 6.23512018301728567233601166870, 7.3025041562910240925792585731, 7.5540482535554403292998634640, 8.47969344821496987810537885588, 9.37218276082817597930340248307, 9.94142237229523806505325157641, 10.62647725652660095790272585240, 11.41184587010420463092457732723, 12.22154665435778011859596558023, 12.952700246665709716639342149812, 13.92436281069372853984260919540, 14.99406295423170273757810883301, 15.22350229512692221601026061980, 16.17528295291466198444885844822, 16.7670270965565653565193162935, 17.60600495483058610141066432640, 18.14764172035183246193059646107, 18.70437187821822989461265326135, 19.630660343738888739443437580937

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