Properties

Label 2-5586-1.1-c1-0-29
Degree $2$
Conductor $5586$
Sign $1$
Analytic cond. $44.6044$
Root an. cond. $6.67865$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 3·5-s + 6-s − 8-s + 9-s − 3·10-s + 3·11-s − 12-s + 4·13-s − 3·15-s + 16-s − 6·17-s − 18-s − 19-s + 3·20-s − 3·22-s − 6·23-s + 24-s + 4·25-s − 4·26-s − 27-s − 3·29-s + 3·30-s + 31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.774·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s + 0.670·20-s − 0.639·22-s − 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.784·26-s − 0.192·27-s − 0.557·29-s + 0.547·30-s + 0.179·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5586\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(44.6044\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.599138289\)
\(L(\frac12)\) \(\approx\) \(1.599138289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
7 \( 1 \)
19 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−8.309621658183829826257937569694, −7.35071840686186553314816658779, −6.48075676791858167917121027514, −6.15905108387705756407606659636, −5.64736859507931332446373433013, −4.47538478154447337214871706894, −3.75264772686247995257940946983, −2.35323352832998832939862950765, −1.79770368897729986401577375142, −0.794050740732270172204497719054, 0.794050740732270172204497719054, 1.79770368897729986401577375142, 2.35323352832998832939862950765, 3.75264772686247995257940946983, 4.47538478154447337214871706894, 5.64736859507931332446373433013, 6.15905108387705756407606659636, 6.48075676791858167917121027514, 7.35071840686186553314816658779, 8.309621658183829826257937569694

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