Properties

Label 2-5445-1.1-c1-0-117
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.38·2-s + 3.66·4-s + 5-s + 2.24·7-s + 3.97·8-s + 2.38·10-s + 5.33·14-s + 2.12·16-s + 0.509·17-s + 2.24·19-s + 3.66·20-s + 4.45·23-s + 25-s + 8.22·28-s − 3.46·29-s − 7.24·31-s − 2.89·32-s + 1.21·34-s + 2.24·35-s + 10.7·37-s + 5.33·38-s + 3.97·40-s + 7.94·41-s + 6.05·43-s + 10.6·46-s + 12.2·47-s − 1.97·49-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.83·4-s + 0.447·5-s + 0.847·7-s + 1.40·8-s + 0.752·10-s + 1.42·14-s + 0.530·16-s + 0.123·17-s + 0.514·19-s + 0.820·20-s + 0.928·23-s + 0.200·25-s + 1.55·28-s − 0.643·29-s − 1.30·31-s − 0.511·32-s + 0.208·34-s + 0.378·35-s + 1.77·37-s + 0.865·38-s + 0.628·40-s + 1.24·41-s + 0.924·43-s + 1.56·46-s + 1.78·47-s − 0.281·49-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: 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

\(L(1)\) \(\approx\) \(7.038923970\)
\(L(\frac12)\) \(\approx\) \(7.038923970\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.38T + 2T^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 0.509T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 7.94T + 41T^{2} \)
43 \( 1 - 6.05T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 3.79T + 53T^{2} \)
59 \( 1 + 4.67T + 59T^{2} \)
61 \( 1 - 7.79T + 61T^{2} \)
67 \( 1 + 8.79T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 6.49T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 0.180T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−7.74899507668353545129820310751, −7.36908135583860990740659797103, −6.46195431943793771862783575614, −5.67501463205552484392428322844, −5.34228806104660944473393694793, −4.48672456022439222054131163585, −3.92341489969317228225152237213, −2.91827259762227594056278687492, −2.25413257360884070837880843258, −1.21299121954803015005032868484, 1.21299121954803015005032868484, 2.25413257360884070837880843258, 2.91827259762227594056278687492, 3.92341489969317228225152237213, 4.48672456022439222054131163585, 5.34228806104660944473393694793, 5.67501463205552484392428322844, 6.46195431943793771862783575614, 7.36908135583860990740659797103, 7.74899507668353545129820310751

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