Properties

Label 2-5586-1.1-c1-0-107
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 + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s − 6·13-s − 15-s + 16-s + 2·17-s + 18-s + 19-s + 20-s − 22-s − 23-s − 24-s − 4·25-s − 6·26-s − 27-s + 4·29-s − 30-s + 32-s + 33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.213·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 1.17·26-s − 0.192·27-s + 0.742·29-s − 0.182·30-s + 0.176·32-s + 0.174·33-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 \( 1 - T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−7.68787319678528296559614269173, −6.85205305716404345521010518949, −6.33093889844065371574278751665, −5.40453040336635734676612220113, −5.05284132474957362416669616688, −4.30072305603804237940284022463, −3.24292617856463510490526901417, −2.43781014545618592878916676974, −1.51514476617454152307994640010, 0, 1.51514476617454152307994640010, 2.43781014545618592878916676974, 3.24292617856463510490526901417, 4.30072305603804237940284022463, 5.05284132474957362416669616688, 5.40453040336635734676612220113, 6.33093889844065371574278751665, 6.85205305716404345521010518949, 7.68787319678528296559614269173

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