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Properties

Label 1-1800-1800.1091-r0-0-0
Degree $1$
Conductor $1800$
Sign $-0.995 + 0.0906i$
Analytic cond. $8.35916$
Root an. cond. $8.35916$
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  + (0.5 + 0.866i)7-s + (−0.669 + 0.743i)11-s + (−0.669 − 0.743i)13-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.978 − 0.207i)23-s + (0.913 + 0.406i)29-s + (−0.913 + 0.406i)31-s + (−0.309 + 0.951i)37-s + (−0.669 − 0.743i)41-s + (−0.5 − 0.866i)43-s + (0.913 + 0.406i)47-s + (−0.5 + 0.866i)49-s + (−0.809 + 0.587i)53-s + (−0.669 − 0.743i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.669 + 0.743i)11-s + (−0.669 − 0.743i)13-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.978 − 0.207i)23-s + (0.913 + 0.406i)29-s + (−0.913 + 0.406i)31-s + (−0.309 + 0.951i)37-s + (−0.669 − 0.743i)41-s + (−0.5 − 0.866i)43-s + (0.913 + 0.406i)47-s + (−0.5 + 0.866i)49-s + (−0.809 + 0.587i)53-s + (−0.669 − 0.743i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.995 + 0.0906i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ -0.995 + 0.0906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01739754109 + 0.3831236049i\)
\(L(\frac12)\) \(\approx\) \(0.01739754109 + 0.3831236049i\)
\(L(1)\) \(\approx\) \(0.8257094905 + 0.1554897976i\)
\(L(1)\) \(\approx\) \(0.8257094905 + 0.1554897976i\)

Euler product

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

−19.839344124698884499282651396746, −19.10487421057960902604293522949, −18.3862571945421596745032680350, −17.65493830771213955278321724083, −16.62079463344247804485660675570, −16.50681754632941921426190352123, −15.414247623420862342628885715312, −14.39906119563860928514665802429, −14.06192140703009418709174307974, −13.25741029424555534867789496265, −12.30928499638824765455428905509, −11.59495595239844924417646171584, −10.77834190619038147284305004494, −10.12435360180246735502345193583, −9.35154437286899742528196981259, −8.19581115578324359254283863999, −7.77856676484567724656096130265, −6.8831006280208094806505103171, −5.95798750998701662669693203655, −5.06955417608973965420592140818, −4.262741691867875567636662577422, −3.44473378230473469572276424796, −2.35773904846213994131948709817, −1.41643470921808759567532336766, −0.12732106848440252055936137749, 1.56009239683931385862518473457, 2.36342121591007539878484319623, 3.16554562500124892335466932972, 4.40657618426062129203555220259, 5.14581503991198723397608260882, 5.78357773620892755579143117849, 6.82904218280574680713490117379, 7.75180810772931240618955826018, 8.34427178407003862194986043618, 9.15886013124772410788970604039, 10.25258652989678355484254312229, 10.543817451446089432988712945014, 11.78491403161783818043813982818, 12.401564567814347971486241171336, 12.86074343827437612793726104276, 14.03138857252534856177244214228, 14.734573921394090688491841738336, 15.382964072385489433490726434264, 15.89792961184351101928131998334, 17.176601864498648128713669244941, 17.49195209033641890122246750625, 18.45031022365826319450409179800, 18.88481966120692117822161416954, 20.002727946382359705701496418080, 20.4083689443480361852186466171

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