Properties

Label 2-5445-1.1-c1-0-11
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  − 1.82·2-s + 1.33·4-s − 5-s + 1.74·7-s + 1.20·8-s + 1.82·10-s − 5.35·13-s − 3.19·14-s − 4.88·16-s − 7.33·17-s + 6.01·19-s − 1.33·20-s − 8.35·23-s + 25-s + 9.78·26-s + 2.33·28-s − 5.53·29-s − 0.0213·31-s + 6.50·32-s + 13.3·34-s − 1.74·35-s − 1.12·37-s − 10.9·38-s − 1.20·40-s + 3.65·41-s − 1.96·43-s + 15.2·46-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.669·4-s − 0.447·5-s + 0.660·7-s + 0.427·8-s + 0.577·10-s − 1.48·13-s − 0.853·14-s − 1.22·16-s − 1.77·17-s + 1.37·19-s − 0.299·20-s − 1.74·23-s + 0.200·25-s + 1.91·26-s + 0.441·28-s − 1.02·29-s − 0.00383·31-s + 1.15·32-s + 2.29·34-s − 0.295·35-s − 0.184·37-s − 1.78·38-s − 0.191·40-s + 0.571·41-s − 0.300·43-s + 2.25·46-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

\(L(1)\) \(\approx\) \(0.4459417575\)
\(L(\frac12)\) \(\approx\) \(0.4459417575\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.82T + 2T^{2} \)
7 \( 1 - 1.74T + 7T^{2} \)
13 \( 1 + 5.35T + 13T^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + 8.35T + 23T^{2} \)
29 \( 1 + 5.53T + 29T^{2} \)
31 \( 1 + 0.0213T + 31T^{2} \)
37 \( 1 + 1.12T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 + 3.81T + 47T^{2} \)
53 \( 1 + 1.67T + 53T^{2} \)
59 \( 1 + 5.66T + 59T^{2} \)
61 \( 1 - 9.27T + 61T^{2} \)
67 \( 1 - 0.350T + 67T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 2.56T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 1.89T + 97T^{2} \)
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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

−8.108895245514226356124484548450, −7.63948035584590490320765833463, −7.15405117234147970819575208702, −6.28007242618763622191049265684, −5.07976331448573435187157462884, −4.62761893519256743056920531612, −3.71129055558268243168103799640, −2.36365885036176747101590122815, −1.79164885240534944179864022575, −0.42584992172214066086248064672, 0.42584992172214066086248064672, 1.79164885240534944179864022575, 2.36365885036176747101590122815, 3.71129055558268243168103799640, 4.62761893519256743056920531612, 5.07976331448573435187157462884, 6.28007242618763622191049265684, 7.15405117234147970819575208702, 7.63948035584590490320765833463, 8.108895245514226356124484548450

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