Properties

Label 2-5390-1.1-c1-0-131
Degree $2$
Conductor $5390$
Sign $-1$
Analytic cond. $43.0393$
Root an. cond. $6.56043$
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 − 5-s + 2·6-s + 8-s + 9-s − 10-s − 11-s + 2·12-s − 2·13-s − 2·15-s + 16-s − 6·17-s + 18-s − 2·19-s − 20-s − 22-s − 6·23-s + 2·24-s + 25-s − 2·26-s − 4·27-s − 2·30-s − 8·31-s + 32-s − 2·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.213·22-s − 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.365·30-s − 1.43·31-s + 0.176·32-s − 0.348·33-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: yes
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.64372537975475166022455310931, −7.35470064658624788519865702495, −6.35277559402025860115369289623, −5.64452750837869894296863372710, −4.57810270451490047924657852732, −4.10411235039070266395550245634, −3.29025930586599630033262076493, −2.45878940537146842264755210381, −1.91644592917315076844868375360, 0, 1.91644592917315076844868375360, 2.45878940537146842264755210381, 3.29025930586599630033262076493, 4.10411235039070266395550245634, 4.57810270451490047924657852732, 5.64452750837869894296863372710, 6.35277559402025860115369289623, 7.35470064658624788519865702495, 7.64372537975475166022455310931

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