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Properties

Label 1-2205-2205.2104-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.391 - 0.920i$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)8-s + (−0.733 − 0.680i)11-s + (0.733 + 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 + 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.826 − 0.563i)26-s + (0.826 + 0.563i)29-s + 31-s + (0.900 − 0.433i)32-s + (−0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)8-s + (−0.733 − 0.680i)11-s + (0.733 + 0.680i)13-s + (0.623 + 0.781i)16-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 + 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.826 − 0.563i)26-s + (0.826 + 0.563i)29-s + 31-s + (0.900 − 0.433i)32-s + (−0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151565524 - 0.7613201000i\)
\(L(\frac12)\) \(\approx\) \(1.151565524 - 0.7613201000i\)
\(L(1)\) \(\approx\) \(0.9013705087 - 0.4762544905i\)
\(L(1)\) \(\approx\) \(0.9013705087 - 0.4762544905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.76276167218494414302091561829, −19.05633734484965957276367134661, −18.10303916347527365301582930887, −17.55072946134855317570247959716, −17.20264764544393646073647422687, −15.88577495004915857228517090969, −15.57861570992154217774984355180, −15.1211333746389739112302228827, −14.0120371552028923301143405797, −13.4189707546609197526475873418, −12.83142023531858107392674093993, −12.09616613117633731759690592782, −10.94478837627804791068855521231, −10.27361848783806157048699729904, −9.308194533312735100161599068220, −8.60281237603876602776624150834, −7.87996317342724632641706878597, −7.18152815599056514844852251389, −6.324762369271000653650925415862, −5.662997511253002666309202841167, −4.76821033672998054052136556320, −4.12582486476754556934949026324, −3.13165946077216937556127676694, −2.13263358846080172059755130823, −0.65083766621132312383232032464, 0.725635116906132237527685156275, 1.762909321748666987430309117349, 2.67154171303200536752086079397, 3.3685345816341414538353585903, 4.4059818154834438228645656910, 4.90201918691512691459670444221, 6.062141998927785718086036605854, 6.578008508493893312746912373132, 8.06900199869039577358489676142, 8.54413133246449300387827492505, 9.31614706930894841656459183863, 10.35650532691229229619705881311, 10.68930241196037127079798833634, 11.64634913647084289971118373671, 12.10700074845572200916471897567, 13.27835766632248024417021479717, 13.441803367966766740768442505042, 14.36118173294799826406733436944, 15.06906598538305615160574809345, 16.110955285493976616027818143014, 16.62411257563674190606127937245, 17.81258010507302179111783425411, 18.31415216702931357318730589802, 18.91630037545818253962961210056, 19.607916409875706265912613625433

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