Properties

Label 2-5445-1.1-c1-0-102
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  − 2.70·2-s + 5.34·4-s − 5-s − 1.07·7-s − 9.04·8-s + 2.70·10-s + 4.34·13-s + 2.92·14-s + 13.8·16-s + 7.75·17-s − 5.26·19-s − 5.34·20-s + 2.15·23-s + 25-s − 11.7·26-s − 5.75·28-s + 1.41·29-s − 4.68·31-s − 19.3·32-s − 21.0·34-s + 1.07·35-s − 2·37-s + 14.2·38-s + 9.04·40-s − 9.41·41-s − 7.60·43-s − 5.84·46-s + ⋯
L(s)  = 1  − 1.91·2-s + 2.67·4-s − 0.447·5-s − 0.407·7-s − 3.19·8-s + 0.856·10-s + 1.20·13-s + 0.780·14-s + 3.45·16-s + 1.88·17-s − 1.20·19-s − 1.19·20-s + 0.449·23-s + 0.200·25-s − 2.30·26-s − 1.08·28-s + 0.263·29-s − 0.840·31-s − 3.42·32-s − 3.60·34-s + 0.182·35-s − 0.328·37-s + 2.31·38-s + 1.43·40-s − 1.47·41-s − 1.15·43-s − 0.861·46-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 \( 1 + 2.70T + 2T^{2} \)
7 \( 1 + 1.07T + 7T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
17 \( 1 - 7.75T + 17T^{2} \)
19 \( 1 + 5.26T + 19T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 + 7.60T + 43T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 + 0.156T + 53T^{2} \)
59 \( 1 + 6.15T + 59T^{2} \)
61 \( 1 - 4.15T + 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 - 4.68T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−8.058209446633277524341945918766, −7.34761624445102672646220656819, −6.57820513206009496686784736963, −6.13255691434261946194397147488, −5.12139338858819136365012137814, −3.56174705688401557978556857401, −3.20443516020982238814556841865, −1.90733479217931900555298096672, −1.12277345345335647661122554878, 0, 1.12277345345335647661122554878, 1.90733479217931900555298096672, 3.20443516020982238814556841865, 3.56174705688401557978556857401, 5.12139338858819136365012137814, 6.13255691434261946194397147488, 6.57820513206009496686784736963, 7.34761624445102672646220656819, 8.058209446633277524341945918766

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