Properties

Label 2-1550-1.1-c1-0-26
Degree $2$
Conductor $1550$
Sign $-1$
Analytic cond. $12.3768$
Root an. cond. $3.51806$
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 − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 2·11-s − 2·12-s + 16-s − 2·17-s − 18-s − 4·19-s − 2·22-s + 4·23-s + 2·24-s + 4·27-s − 4·29-s − 31-s − 32-s − 4·33-s + 2·34-s + 36-s + 8·37-s + 4·38-s + 6·41-s − 2·43-s + 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.426·22-s + 0.834·23-s + 0.408·24-s + 0.769·27-s − 0.742·29-s − 0.179·31-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + 1.31·37-s + 0.648·38-s + 0.937·41-s − 0.304·43-s + 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1550\)    =    \(2 \cdot 5^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(12.3768\)
Root analytic conductor: \(3.51806\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1550,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
31 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.108228111732168721022987729500, −8.349740398714681621706993236924, −7.34656286915359393054915647417, −6.52577117551676169718812845188, −6.01571451188682654493203805347, −5.02312771424614556239474530161, −4.06198490562093620006357244819, −2.67517211764250224916899504851, −1.32334138024336779760554246413, 0, 1.32334138024336779760554246413, 2.67517211764250224916899504851, 4.06198490562093620006357244819, 5.02312771424614556239474530161, 6.01571451188682654493203805347, 6.52577117551676169718812845188, 7.34656286915359393054915647417, 8.349740398714681621706993236924, 9.108228111732168721022987729500

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