Properties

Label 2-5445-1.1-c1-0-109
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  − 2.39·2-s + 3.73·4-s − 5-s + 1.96·7-s − 4.15·8-s + 2.39·10-s + 2.92·13-s − 4.71·14-s + 2.48·16-s − 6.62·17-s − 3.56·19-s − 3.73·20-s + 2.78·23-s + 25-s − 6.99·26-s + 7.35·28-s − 3.51·29-s + 8.11·31-s + 2.36·32-s + 15.8·34-s − 1.96·35-s + 4.42·37-s + 8.52·38-s + 4.15·40-s + 7.07·41-s − 10.8·43-s − 6.67·46-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.86·4-s − 0.447·5-s + 0.744·7-s − 1.46·8-s + 0.757·10-s + 0.809·13-s − 1.26·14-s + 0.620·16-s − 1.60·17-s − 0.816·19-s − 0.835·20-s + 0.581·23-s + 0.200·25-s − 1.37·26-s + 1.39·28-s − 0.653·29-s + 1.45·31-s + 0.418·32-s + 2.71·34-s − 0.332·35-s + 0.728·37-s + 1.38·38-s + 0.657·40-s + 1.10·41-s − 1.66·43-s − 0.984·46-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 \( 1 + 2.39T + 2T^{2} \)
7 \( 1 - 1.96T + 7T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 + 6.62T + 17T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 + 3.51T + 29T^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 - 4.42T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 0.0271T + 47T^{2} \)
53 \( 1 - 4.10T + 53T^{2} \)
59 \( 1 + 9.51T + 59T^{2} \)
61 \( 1 - 3.10T + 61T^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 + 6.05T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 5.04T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.159558952483331567522780456908, −7.30397180239655040881321901887, −6.65789267668518395239490162765, −6.05389160238933267348974040908, −4.75720423973631311573088215106, −4.16861104553604226795457099427, −2.88054241955693325784221240176, −2.00622066977541151365099605475, −1.15388943809720797810965496748, 0, 1.15388943809720797810965496748, 2.00622066977541151365099605475, 2.88054241955693325784221240176, 4.16861104553604226795457099427, 4.75720423973631311573088215106, 6.05389160238933267348974040908, 6.65789267668518395239490162765, 7.30397180239655040881321901887, 8.159558952483331567522780456908

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