Properties

Label 2-5390-1.1-c1-0-103
Degree $2$
Conductor $5390$
Sign $-1$
Analytic cond. $43.0393$
Root an. cond. $6.56043$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.73·3-s + 4-s + 5-s − 2.73·6-s + 8-s + 4.46·9-s + 10-s + 11-s − 2.73·12-s + 1.46·13-s − 2.73·15-s + 16-s + 3.46·17-s + 4.46·18-s − 6.73·19-s + 20-s + 22-s − 8.19·23-s − 2.73·24-s + 25-s + 1.46·26-s − 3.99·27-s − 4.73·29-s − 2.73·30-s − 2·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.57·3-s + 0.5·4-s + 0.447·5-s − 1.11·6-s + 0.353·8-s + 1.48·9-s + 0.316·10-s + 0.301·11-s − 0.788·12-s + 0.406·13-s − 0.705·15-s + 0.250·16-s + 0.840·17-s + 1.05·18-s − 1.54·19-s + 0.223·20-s + 0.213·22-s − 1.70·23-s − 0.557·24-s + 0.200·25-s + 0.287·26-s − 0.769·27-s − 0.878·29-s − 0.498·30-s − 0.359·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5390\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(43.0393\)
Root analytic conductor: \(6.56043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5390} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5390,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 2.73T + 3T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 0.732T + 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 7.26T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−7.62057715415451008374069364640, −6.62464185759589633863130046865, −6.23224410377974933078316324515, −5.73218285338477303861636104920, −5.05348393536037774514201049731, −4.25436906620266612870053618325, −3.60436574941212801996809949438, −2.22132572776758161169944011955, −1.36902028943590871257689805309, 0, 1.36902028943590871257689805309, 2.22132572776758161169944011955, 3.60436574941212801996809949438, 4.25436906620266612870053618325, 5.05348393536037774514201049731, 5.73218285338477303861636104920, 6.23224410377974933078316324515, 6.62464185759589633863130046865, 7.62057715415451008374069364640

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