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Properties

Label 2-1800-1800.691-c0-0-0
Degree $2$
Conductor $1800$
Sign $0.946 - 0.322i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
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  + (0.406 + 0.913i)2-s + (−0.309 − 0.951i)3-s + (−0.669 + 0.743i)4-s + (0.743 + 0.669i)5-s + (0.743 − 0.669i)6-s + (−0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (1.47 − 0.658i)11-s + (0.913 + 0.406i)12-s + (0.251 − 0.564i)13-s + (0.104 − 0.994i)14-s + (0.406 − 0.913i)15-s + (−0.104 − 0.994i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.309 − 0.951i)3-s + (−0.669 + 0.743i)4-s + (0.743 + 0.669i)5-s + (0.743 − 0.669i)6-s + (−0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (1.47 − 0.658i)11-s + (0.913 + 0.406i)12-s + (0.251 − 0.564i)13-s + (0.104 − 0.994i)14-s + (0.406 − 0.913i)15-s + (−0.104 − 0.994i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.946 - 0.322i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.946 - 0.322i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279020142\)
\(L(\frac12)\) \(\approx\) \(1.279020142\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.406 - 0.913i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
good7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \)
13 \( 1 + (-0.251 + 0.564i)T + (-0.669 - 0.743i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \)
29 \( 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)T^{2} \)
31 \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.336 + 1.58i)T + (-0.913 - 0.406i)T^{2} \)
53 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.669 + 0.743i)T^{2} \)
61 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
67 \( 1 + (-0.604 + 0.128i)T + (0.913 - 0.406i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.336 + 1.58i)T + (-0.913 - 0.406i)T^{2} \)
83 \( 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.913 + 0.406i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.154972138092749423554009219035, −8.780474435281713059097792895836, −7.45716990592763097488693198417, −6.85936417998236597744854326003, −6.59120145185288736536297392710, −5.71568929047272334128660240038, −4.97712402832878895119092956980, −3.37453817905252439252046660163, −2.99250640731686283080118512550, −1.07761815183592965076661628003, 1.34726965669172525472425233485, 2.53214035042432769258915121835, 3.84104475217453178506036654742, 4.17154579132512306058050801008, 5.27714675435774555373011803747, 6.06501705629692821253257555184, 6.43513728706203302761195495469, 8.338876049865749857760635711265, 9.155512650694377993239298961010, 9.687139038759259613001505659235

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