Properties

Label 2-5445-1.1-c1-0-115
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  − 2·4-s + 5-s − 1.73·7-s + 4·16-s − 3.46·17-s + 5.19·19-s − 2·20-s − 6·23-s + 25-s + 3.46·28-s + 6.92·29-s + 31-s − 1.73·35-s − 5·37-s + 3.46·41-s − 10.3·43-s + 12·47-s − 4·49-s − 6·53-s + 12.1·61-s − 8·64-s − 5·67-s + 6.92·68-s + 6·71-s + 1.73·73-s − 10.3·76-s + 15.5·79-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.654·7-s + 16-s − 0.840·17-s + 1.19·19-s − 0.447·20-s − 1.25·23-s + 0.200·25-s + 0.654·28-s + 1.28·29-s + 0.179·31-s − 0.292·35-s − 0.821·37-s + 0.541·41-s − 1.58·43-s + 1.75·47-s − 0.571·49-s − 0.824·53-s + 1.55·61-s − 64-s − 0.610·67-s + 0.840·68-s + 0.712·71-s + 0.202·73-s − 1.19·76-s + 1.75·79-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 \( 1 + 2T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 5.19T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−8.030411022038040963935798238524, −6.99250165850645537751584055318, −6.36424756758133174761541488458, −5.56132720129851857202182063052, −4.93858295195974718194769374186, −4.09624654899216395589804544139, −3.36617818308706827410639026401, −2.43500839787088700842339615987, −1.19494010893566666338565529033, 0, 1.19494010893566666338565529033, 2.43500839787088700842339615987, 3.36617818308706827410639026401, 4.09624654899216395589804544139, 4.93858295195974718194769374186, 5.56132720129851857202182063052, 6.36424756758133174761541488458, 6.99250165850645537751584055318, 8.030411022038040963935798238524

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