Properties

Label 2-5445-1.1-c1-0-104
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  + 0.741·2-s − 1.44·4-s − 5-s − 3.30·7-s − 2.55·8-s − 0.741·10-s + 1.81·13-s − 2.44·14-s + 1.00·16-s + 3.30·17-s − 1.48·19-s + 1.44·20-s + 4.89·23-s + 25-s + 1.34·26-s + 4.78·28-s + 8.08·29-s − 2·31-s + 5.86·32-s + 2.44·34-s + 3.30·35-s + 2.89·37-s − 1.10·38-s + 2.55·40-s − 5.11·41-s + 3.30·43-s + 3.63·46-s + ⋯
L(s)  = 1  + 0.524·2-s − 0.724·4-s − 0.447·5-s − 1.24·7-s − 0.904·8-s − 0.234·10-s + 0.504·13-s − 0.654·14-s + 0.250·16-s + 0.800·17-s − 0.340·19-s + 0.324·20-s + 1.02·23-s + 0.200·25-s + 0.264·26-s + 0.904·28-s + 1.50·29-s − 0.359·31-s + 1.03·32-s + 0.420·34-s + 0.558·35-s + 0.476·37-s − 0.178·38-s + 0.404·40-s − 0.799·41-s + 0.503·43-s + 0.535·46-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 \( 1 - 0.741T + 2T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 + 1.48T + 19T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 - 8.08T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 2.89T + 37T^{2} \)
41 \( 1 + 5.11T + 41T^{2} \)
43 \( 1 - 3.30T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 + 1.10T + 59T^{2} \)
61 \( 1 - 9.57T + 61T^{2} \)
67 \( 1 + 0.898T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 + 4.78T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−7.86725503529945546445740315309, −6.91250396033923775360842178126, −6.32120269292809431133693154448, −5.61705980175067973614507497083, −4.80224827357890861900771394492, −4.07790753065333851054698253189, −3.27396086203878247747772280890, −2.88606703393221000419150491996, −1.12547456000065746731045234235, 0, 1.12547456000065746731045234235, 2.88606703393221000419150491996, 3.27396086203878247747772280890, 4.07790753065333851054698253189, 4.80224827357890861900771394492, 5.61705980175067973614507497083, 6.32120269292809431133693154448, 6.91250396033923775360842178126, 7.86725503529945546445740315309

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