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Properties

Label 1-1800-1800.1787-r1-0-0
Degree $1$
Conductor $1800$
Sign $0.709 + 0.704i$
Analytic cond. $193.436$
Root an. cond. $193.436$
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  + (0.866 + 0.5i)7-s + (0.978 + 0.207i)11-s + (−0.207 − 0.978i)13-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.743 − 0.669i)23-s + (0.104 + 0.994i)29-s + (0.104 − 0.994i)31-s + (−0.951 − 0.309i)37-s + (0.978 − 0.207i)41-s + (0.866 + 0.5i)43-s + (0.994 − 0.104i)47-s + (0.5 + 0.866i)49-s + (0.587 + 0.809i)53-s + (−0.978 + 0.207i)59-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.978 + 0.207i)11-s + (−0.207 − 0.978i)13-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.743 − 0.669i)23-s + (0.104 + 0.994i)29-s + (0.104 − 0.994i)31-s + (−0.951 − 0.309i)37-s + (0.978 − 0.207i)41-s + (0.866 + 0.5i)43-s + (0.994 − 0.104i)47-s + (0.5 + 0.866i)49-s + (0.587 + 0.809i)53-s + (−0.978 + 0.207i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ 0.709 + 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.500506869 + 1.030634559i\)
\(L(\frac12)\) \(\approx\) \(2.500506869 + 1.030634559i\)
\(L(1)\) \(\approx\) \(1.277956219 + 0.1408678381i\)
\(L(1)\) \(\approx\) \(1.277956219 + 0.1408678381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.207 - 0.978i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.994 - 0.104i)T \)
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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

−19.84313184600410350145397685447, −19.359353446360955091338689477628, −18.363660296952870442214085445438, −17.581821488971268596548896402776, −17.176294962156885148320647205283, −16.18979801828878615625802102853, −15.60422921846871161302564357812, −14.53954533885715098530826754204, −13.90641129622525721639397866949, −13.6195189873889880769145230283, −12.20506106715323293896116720351, −11.62877189353286508948509715841, −11.14107206643820959186348858575, −10.09148865444293032408671179661, −9.24212752420120821727587741651, −8.68412552253545756516872985475, −7.54543007195960865864227814714, −7.05237079277166855461405473003, −6.13530045273967567193672348686, −5.067005060551829238931668011044, −4.370738838469432115051910532849, −3.62497614853356823273236954345, −2.39053253769079569756479444928, −1.51068717137758369099015692070, −0.59049231461649915672795832039, 0.86522735717679564526586392110, 1.7764402836378949332072294049, 2.65106199976120529333049156755, 3.811562274978292805371484229781, 4.50505462264151207272408749541, 5.5599948148612099696059539796, 6.088842334456013821865893720021, 7.23633419482678681851726122438, 7.95722148030668007064090965931, 8.72333219058755697998056925033, 9.431663673624841456769484407481, 10.476958623095001002009096157116, 11.0204939919440629279029322434, 12.19081211241046252687244963338, 12.28101876611213809319077350164, 13.48992418727429968580543550907, 14.36621730265882406600101343698, 14.82338808803059067014004299752, 15.58893523678563753303273063830, 16.43274503661827399514564158311, 17.38994501767177291459133913495, 17.77867490310664736429789565294, 18.55001319931340777077747591116, 19.425470163426077805858731966613, 20.20237352762554097559443232936

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