Properties

Label 2-5445-1.1-c1-0-76
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  − 0.193·2-s − 1.96·4-s − 5-s − 3.35·7-s + 0.768·8-s + 0.193·10-s − 2.96·13-s + 0.649·14-s + 3.77·16-s − 4.57·17-s + 4.31·19-s + 1.96·20-s + 6.70·23-s + 25-s + 0.574·26-s + 6.57·28-s − 3.61·29-s + 9.92·31-s − 2.26·32-s + 0.887·34-s + 3.35·35-s − 2·37-s − 0.836·38-s − 0.768·40-s − 4.38·41-s + 9.27·43-s − 1.29·46-s + ⋯
L(s)  = 1  − 0.137·2-s − 0.981·4-s − 0.447·5-s − 1.26·7-s + 0.271·8-s + 0.0613·10-s − 0.821·13-s + 0.173·14-s + 0.943·16-s − 1.10·17-s + 0.989·19-s + 0.438·20-s + 1.39·23-s + 0.200·25-s + 0.112·26-s + 1.24·28-s − 0.670·29-s + 1.78·31-s − 0.401·32-s + 0.152·34-s + 0.566·35-s − 0.328·37-s − 0.135·38-s − 0.121·40-s − 0.685·41-s + 1.41·43-s − 0.191·46-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 \( 1 + 0.193T + 2T^{2} \)
7 \( 1 + 3.35T + 7T^{2} \)
13 \( 1 + 2.96T + 13T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 - 4.31T + 19T^{2} \)
23 \( 1 - 6.70T + 23T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 - 9.92T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 4.38T + 41T^{2} \)
43 \( 1 - 9.27T + 43T^{2} \)
47 \( 1 - 9.92T + 47T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 - 5.92T + 67T^{2} \)
71 \( 1 + 9.92T + 71T^{2} \)
73 \( 1 - 7.73T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 2.77T + 89T^{2} \)
97 \( 1 - 0.0752T + 97T^{2} \)
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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

−7.79095076085778356567920473831, −7.14313796504668757166890309028, −6.49581206103246332263642962859, −5.54271166514150688890944391012, −4.80961646063123247639063662300, −4.14288684155752306807239951676, −3.27942111342585387335074407755, −2.60631314058696451192810523850, −0.968584791953022547473595614955, 0, 0.968584791953022547473595614955, 2.60631314058696451192810523850, 3.27942111342585387335074407755, 4.14288684155752306807239951676, 4.80961646063123247639063662300, 5.54271166514150688890944391012, 6.49581206103246332263642962859, 7.14313796504668757166890309028, 7.79095076085778356567920473831

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