Properties

Degree $2$
Conductor $6069$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 2·5-s + 6-s + 7-s + 3·8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s − 14-s − 2·15-s − 16-s − 18-s + 4·19-s − 2·20-s − 21-s + 4·22-s − 3·24-s − 25-s + 2·26-s − 27-s − 28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6069\)    =    \(3 \cdot 7 \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6069} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6069,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8752924544\)
\(L(\frac12)\) \(\approx\) \(0.8752924544\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.150173736744291796300091652744, −7.43276089750609294899404044756, −6.83652065925235047357009366271, −5.70635600713953756001504269408, −5.24463571202663205176745275817, −4.77630342481941975935989178304, −3.69050895952500294993723894736, −2.48232080233279175904585277989, −1.65235155148622815444498272942, −0.57807668787933325619389723813, 0.57807668787933325619389723813, 1.65235155148622815444498272942, 2.48232080233279175904585277989, 3.69050895952500294993723894736, 4.77630342481941975935989178304, 5.24463571202663205176745275817, 5.70635600713953756001504269408, 6.83652065925235047357009366271, 7.43276089750609294899404044756, 8.150173736744291796300091652744

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