Properties

Label 2-5445-1.1-c1-0-122
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  − 1.47·2-s + 0.182·4-s − 5-s + 2.29·7-s + 2.68·8-s + 1.47·10-s + 2.14·13-s − 3.39·14-s − 4.33·16-s + 0.544·17-s + 2.18·19-s − 0.182·20-s − 2.03·23-s + 25-s − 3.16·26-s + 0.418·28-s − 9.94·29-s + 6.77·31-s + 1.02·32-s − 0.804·34-s − 2.29·35-s − 8.81·37-s − 3.22·38-s − 2.68·40-s − 1.82·41-s + 0.620·43-s + 3.01·46-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.0911·4-s − 0.447·5-s + 0.867·7-s + 0.949·8-s + 0.467·10-s + 0.593·13-s − 0.906·14-s − 1.08·16-s + 0.132·17-s + 0.500·19-s − 0.0407·20-s − 0.425·23-s + 0.200·25-s − 0.620·26-s + 0.0790·28-s − 1.84·29-s + 1.21·31-s + 0.181·32-s − 0.137·34-s − 0.387·35-s − 1.44·37-s − 0.522·38-s − 0.424·40-s − 0.284·41-s + 0.0946·43-s + 0.444·46-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 \( 1 + 1.47T + 2T^{2} \)
7 \( 1 - 2.29T + 7T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 - 0.544T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 + 9.94T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 + 8.81T + 37T^{2} \)
41 \( 1 + 1.82T + 41T^{2} \)
43 \( 1 - 0.620T + 43T^{2} \)
47 \( 1 - 0.378T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 8.07T + 59T^{2} \)
61 \( 1 + 8.72T + 61T^{2} \)
67 \( 1 + 9.75T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + 7.85T + 73T^{2} \)
79 \( 1 - 9.22T + 79T^{2} \)
83 \( 1 + 9.45T + 83T^{2} \)
89 \( 1 + 0.583T + 89T^{2} \)
97 \( 1 + 5.37T + 97T^{2} \)
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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

−7.959704767506744893810750127392, −7.43953111743902847367355785813, −6.63810909966374037572873007696, −5.59466555401448692156068221026, −4.86543543139863851285295400014, −4.12593547986538733984048277707, −3.29240755448038010548326197758, −1.93159533831963081943074369533, −1.23564031931181568632074694770, 0, 1.23564031931181568632074694770, 1.93159533831963081943074369533, 3.29240755448038010548326197758, 4.12593547986538733984048277707, 4.86543543139863851285295400014, 5.59466555401448692156068221026, 6.63810909966374037572873007696, 7.43953111743902847367355785813, 7.959704767506744893810750127392

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