Properties

Label 2-5445-1.1-c1-0-134
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  − 2.45·2-s + 4.03·4-s + 5-s + 3.28·7-s − 4.98·8-s − 2.45·10-s − 0.313·13-s − 8.07·14-s + 4.18·16-s − 5·17-s − 7.45·19-s + 4.03·20-s − 1.07·23-s + 25-s + 0.769·26-s + 13.2·28-s + 5.03·29-s + 3.44·31-s − 0.310·32-s + 12.2·34-s + 3.28·35-s + 2.63·37-s + 18.3·38-s − 4.98·40-s − 10.8·41-s − 5.51·43-s + 2.63·46-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.01·4-s + 0.447·5-s + 1.24·7-s − 1.76·8-s − 0.776·10-s − 0.0868·13-s − 2.15·14-s + 1.04·16-s − 1.21·17-s − 1.71·19-s + 0.901·20-s − 0.223·23-s + 0.200·25-s + 0.150·26-s + 2.50·28-s + 0.935·29-s + 0.619·31-s − 0.0549·32-s + 2.10·34-s + 0.555·35-s + 0.433·37-s + 2.96·38-s − 0.788·40-s − 1.69·41-s − 0.840·43-s + 0.388·46-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 \( 1 + 2.45T + 2T^{2} \)
7 \( 1 - 3.28T + 7T^{2} \)
13 \( 1 + 0.313T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 7.45T + 19T^{2} \)
23 \( 1 + 1.07T + 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 - 3.44T + 31T^{2} \)
37 \( 1 - 2.63T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 4.93T + 53T^{2} \)
59 \( 1 - 9.16T + 59T^{2} \)
61 \( 1 - 9.18T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + 3.07T + 71T^{2} \)
73 \( 1 + 8.65T + 73T^{2} \)
79 \( 1 - 5.41T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 - 0.224T + 97T^{2} \)
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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

−8.224133392959007855140785165089, −7.22521899846432655121242746421, −6.67053237369217928900058896661, −5.99568721121209132141053936442, −4.86452796384073244747523793362, −4.22715613916839593544633580713, −2.64299161682687816546766972541, −2.03993546028406896064462892073, −1.30139408051585678904498526671, 0, 1.30139408051585678904498526671, 2.03993546028406896064462892073, 2.64299161682687816546766972541, 4.22715613916839593544633580713, 4.86452796384073244747523793362, 5.99568721121209132141053936442, 6.67053237369217928900058896661, 7.22521899846432655121242746421, 8.224133392959007855140785165089

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