Properties

Label 2-5586-1.1-c1-0-109
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 4·11-s + 12-s + 16-s + 2·17-s − 18-s − 19-s − 4·22-s − 2·23-s − 24-s − 5·25-s + 27-s − 6·29-s − 6·31-s − 32-s + 4·33-s − 2·34-s + 36-s − 8·37-s + 38-s − 10·41-s − 12·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.852·22-s − 0.417·23-s − 0.204·24-s − 25-s + 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s − 1.31·37-s + 0.162·38-s − 1.56·41-s − 1.82·43-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: \(1\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.963827187419789211430327684788, −7.12214234772244724841434492496, −6.63116441718754640169590116970, −5.77047250595078190707289552234, −4.88911068840059762819121883590, −3.63810961775377186651852118635, −3.47720922400606981079861592555, −1.98418461348406802916978160680, −1.56373695223205832372938931343, 0, 1.56373695223205832372938931343, 1.98418461348406802916978160680, 3.47720922400606981079861592555, 3.63810961775377186651852118635, 4.88911068840059762819121883590, 5.77047250595078190707289552234, 6.63116441718754640169590116970, 7.12214234772244724841434492496, 7.963827187419789211430327684788

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