Properties

Label 2-5445-1.1-c1-0-131
Degree $2$
Conductor $5445$
Sign $-1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
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  − 1.73·2-s + 0.999·4-s + 5-s + 1.73·8-s − 1.73·10-s + 3.46·13-s − 5·16-s − 3.46·19-s + 0.999·20-s + 25-s − 5.99·26-s + 3.46·29-s − 8·31-s + 5.19·32-s − 2·37-s + 5.99·38-s + 1.73·40-s + 10.3·41-s − 6.92·43-s − 7·49-s − 1.73·50-s + 3.46·52-s + 6·53-s − 5.99·58-s − 12·59-s − 13.8·61-s + 13.8·62-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s + 0.447·5-s + 0.612·8-s − 0.547·10-s + 0.960·13-s − 1.25·16-s − 0.794·19-s + 0.223·20-s + 0.200·25-s − 1.17·26-s + 0.643·29-s − 1.43·31-s + 0.918·32-s − 0.328·37-s + 0.973·38-s + 0.273·40-s + 1.62·41-s − 1.05·43-s − 49-s − 0.244·50-s + 0.480·52-s + 0.824·53-s − 0.787·58-s − 1.56·59-s − 1.77·61-s + 1.75·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(43.4785\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5445} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + 1.73T + 2T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 3.46T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.906328144107939052519515140733, −7.34958878057739344390104621086, −6.45617804208935848704043645052, −5.90772676532914122421201284167, −4.86571355766844155665821477428, −4.13150893752785575554679181577, −3.08966285964689345249438516649, −1.95764131220924322066085851460, −1.27098393232889778767349150151, 0, 1.27098393232889778767349150151, 1.95764131220924322066085851460, 3.08966285964689345249438516649, 4.13150893752785575554679181577, 4.86571355766844155665821477428, 5.90772676532914122421201284167, 6.45617804208935848704043645052, 7.34958878057739344390104621086, 7.906328144107939052519515140733

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