Properties

Label 2-5445-1.1-c1-0-135
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.62·2-s + 4.88·4-s − 5-s + 2.62·7-s + 7.56·8-s − 2.62·10-s + 6.72·13-s + 6.88·14-s + 10.0·16-s + 1.78·17-s + 1.48·19-s − 4.88·20-s − 5.20·23-s + 25-s + 17.6·26-s + 12.8·28-s + 1.17·29-s − 4.56·31-s + 11.3·32-s + 4.67·34-s − 2.62·35-s − 0.679·37-s + 3.88·38-s − 7.56·40-s + 5.49·41-s − 3.80·43-s − 13.6·46-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.44·4-s − 0.447·5-s + 0.991·7-s + 2.67·8-s − 0.829·10-s + 1.86·13-s + 1.83·14-s + 2.52·16-s + 0.432·17-s + 0.339·19-s − 1.09·20-s − 1.08·23-s + 0.200·25-s + 3.46·26-s + 2.42·28-s + 0.218·29-s − 0.819·31-s + 2.00·32-s + 0.802·34-s − 0.443·35-s − 0.111·37-s + 0.630·38-s − 1.19·40-s + 0.858·41-s − 0.579·43-s − 2.01·46-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: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.843482059\)
\(L(\frac12)\) \(\approx\) \(7.843482059\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.62T + 2T^{2} \)
7 \( 1 - 2.62T + 7T^{2} \)
13 \( 1 - 6.72T + 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 + 5.20T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + 0.679T + 37T^{2} \)
41 \( 1 - 5.49T + 41T^{2} \)
43 \( 1 + 3.80T + 43T^{2} \)
47 \( 1 + 6.66T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 - 3.88T + 59T^{2} \)
61 \( 1 + 3.21T + 61T^{2} \)
67 \( 1 - 4.66T + 67T^{2} \)
71 \( 1 + 3.88T + 71T^{2} \)
73 \( 1 - 3.56T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 7.33T + 83T^{2} \)
89 \( 1 + 3.44T + 89T^{2} \)
97 \( 1 - 1.47T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.983481222095406973695500897790, −7.31594926732008572454215018133, −6.34143165378978095145765728152, −5.95751169678947598994812806906, −5.09937456308576003552914070798, −4.53379037827791397236438205351, −3.66762349772681545755249805536, −3.34756516447430458336589498650, −2.07181980091514707276226638788, −1.30779423879178471214553783249, 1.30779423879178471214553783249, 2.07181980091514707276226638788, 3.34756516447430458336589498650, 3.66762349772681545755249805536, 4.53379037827791397236438205351, 5.09937456308576003552914070798, 5.95751169678947598994812806906, 6.34143165378978095145765728152, 7.31594926732008572454215018133, 7.983481222095406973695500897790

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