Properties

Label 2-5445-1.1-c1-0-116
Degree $2$
Conductor $5445$
Sign $-1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
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 − 4-s − 5-s − 3·7-s − 3·8-s − 10-s + 4·13-s − 3·14-s − 16-s + 4·19-s + 20-s + 8·23-s + 25-s + 4·26-s + 3·28-s − 6·29-s − 2·31-s + 5·32-s + 3·35-s − 8·37-s + 4·38-s + 3·40-s + 5·41-s + 5·43-s + 8·46-s + 3·47-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s − 1.06·8-s − 0.316·10-s + 1.10·13-s − 0.801·14-s − 1/4·16-s + 0.917·19-s + 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.566·28-s − 1.11·29-s − 0.359·31-s + 0.883·32-s + 0.507·35-s − 1.31·37-s + 0.648·38-s + 0.474·40-s + 0.780·41-s + 0.762·43-s + 1.17·46-s + 0.437·47-s + 2/7·49-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: yes
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 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−7.67335182434830898376879844080, −7.03047752962693739280456902249, −6.18645919576467483734066695447, −5.63502186312600170822729502840, −4.86776974957718181629681238758, −3.96362252317764299608561678448, −3.38823819590339762608687427091, −2.87823196713130212091623991830, −1.20143188188281715606737689033, 0, 1.20143188188281715606737689033, 2.87823196713130212091623991830, 3.38823819590339762608687427091, 3.96362252317764299608561678448, 4.86776974957718181629681238758, 5.63502186312600170822729502840, 6.18645919576467483734066695447, 7.03047752962693739280456902249, 7.67335182434830898376879844080

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