↑

Properties

Label 1-2205-2205.2074-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

Related objects

Downloads

Learn more

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: $\chi_{2205} (2074, \cdot )$
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 \( 1 + (0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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