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Properties

Label 1-1800-1800.1429-r0-0-0
Degree $1$
Conductor $1800$
Sign $-0.282 + 0.959i$
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 + (0.104 − 0.994i)11-s + (−0.104 − 0.994i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.913 + 0.406i)23-s + (−0.669 + 0.743i)29-s + (0.669 + 0.743i)31-s + (−0.809 + 0.587i)37-s + (−0.104 − 0.994i)41-s + (−0.5 − 0.866i)43-s + (−0.669 + 0.743i)47-s + (−0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (0.104 − 0.994i)11-s + (−0.104 − 0.994i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.913 + 0.406i)23-s + (−0.669 + 0.743i)29-s + (0.669 + 0.743i)31-s + (−0.809 + 0.587i)37-s + (−0.104 − 0.994i)41-s + (−0.5 − 0.866i)43-s + (−0.669 + 0.743i)47-s + (−0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.104 + 0.994i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6474472099 + 0.8654605483i\)
\(L(\frac12)\) \(\approx\) \(0.6474472099 + 0.8654605483i\)
\(L(1)\) \(\approx\) \(0.9609832084 + 0.1867071002i\)
\(L(1)\) \(\approx\) \(0.9609832084 + 0.1867071002i\)

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 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−19.90347568076653546862955166140, −19.479616516923552561364586996539, −18.291225803298042448643567914378, −17.85488752928370315893423292032, −16.99117302977002067658745435217, −16.461821337470664667919470683662, −15.475902584122998217179386594040, −14.78957420901793145604962796361, −13.971267651087244404092341668661, −13.453319538482721329894484063283, −12.52742575645871839486065755682, −11.51961606006665810469027293596, −11.23416257292838381911861876619, −9.97971396407723552406374017865, −9.62473429649249456291317913019, −8.56433386024236596517335873702, −7.68935983209892419490725358265, −6.968920776486334580971638815379, −6.38742273976709793385908895667, −4.96159837833518513893699838151, −4.52320298079568078716725958126, −3.72934340871087336278099678893, −2.37518099990614482854030820718, −1.7445538963427194408974551386, −0.375673930595275847664465769, 1.27812485997042618912416988521, 2.14764156900361466243790405741, 3.212222830746104557366737843945, 3.92114472450991931029610353029, 5.18115738635750924872538772728, 5.71828251976588479336379211506, 6.43737142443464622018414457677, 7.65534770923331856946534099488, 8.432756471003110516970891810322, 8.75753002672412386070518090058, 10.01896932765680509211407332948, 10.63352068847175074370676050360, 11.45743206195825928563474383136, 12.26720589037999455939451347275, 12.82192286181975952869062415680, 13.87305998912855220890421306524, 14.46306937931465011358484898249, 15.368393722446384455154486627046, 15.790773435927583835608776393302, 16.847540959626665583730754739, 17.46844648941233233469073081556, 18.28281731167104926833423673063, 18.88620242510448352140748432577, 19.61215402271998846143247822736, 20.45683653220508102038112740162

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