Properties

Label 2-5445-1.1-c1-0-129
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  + 0.772·2-s − 1.40·4-s − 5-s + 1.62·7-s − 2.62·8-s − 0.772·10-s − 4.80·13-s + 1.25·14-s + 0.772·16-s + 2.17·17-s + 5.17·19-s + 1.40·20-s − 4.62·23-s + 25-s − 3.71·26-s − 2.28·28-s + 2.45·29-s + 5.80·31-s + 5.85·32-s + 1.68·34-s − 1.62·35-s + 8.88·37-s + 4·38-s + 2.62·40-s − 5.09·41-s − 1.54·43-s − 3.57·46-s + ⋯
L(s)  = 1  + 0.546·2-s − 0.701·4-s − 0.447·5-s + 0.616·7-s − 0.929·8-s − 0.244·10-s − 1.33·13-s + 0.336·14-s + 0.193·16-s + 0.527·17-s + 1.18·19-s + 0.313·20-s − 0.965·23-s + 0.200·25-s − 0.728·26-s − 0.432·28-s + 0.455·29-s + 1.04·31-s + 1.03·32-s + 0.288·34-s − 0.275·35-s + 1.46·37-s + 0.648·38-s + 0.415·40-s − 0.795·41-s − 0.235·43-s − 0.527·46-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 \( 1 - 0.772T + 2T^{2} \)
7 \( 1 - 1.62T + 7T^{2} \)
13 \( 1 + 4.80T + 13T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 - 5.17T + 19T^{2} \)
23 \( 1 + 4.62T + 23T^{2} \)
29 \( 1 - 2.45T + 29T^{2} \)
31 \( 1 - 5.80T + 31T^{2} \)
37 \( 1 - 8.88T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 + 1.54T + 43T^{2} \)
47 \( 1 + 9.72T + 47T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 - 7.15T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 4.72T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 - 4.89T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 4.72T + 97T^{2} \)
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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

−7.979682407620983258205508023123, −7.18311412156709596113941036948, −6.22051409707319989732536995794, −5.41813135528031197441433374113, −4.78267052133339397383141228919, −4.32716532626397387796716312982, −3.33068655945813211326110106340, −2.64584457251962317860744735760, −1.26407724166675357876117583778, 0, 1.26407724166675357876117583778, 2.64584457251962317860744735760, 3.33068655945813211326110106340, 4.32716532626397387796716312982, 4.78267052133339397383141228919, 5.41813135528031197441433374113, 6.22051409707319989732536995794, 7.18311412156709596113941036948, 7.979682407620983258205508023123

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