Properties

Label 2-5148-1.1-c1-0-30
Degree $2$
Conductor $5148$
Sign $-1$
Analytic cond. $41.1069$
Root an. cond. $6.41147$
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  − 4.38·5-s + 3.21·7-s + 11-s + 13-s + 1.86·17-s − 6.18·19-s − 3.48·23-s + 14.2·25-s − 2.83·29-s − 1.08·31-s − 14.0·35-s − 2.69·37-s + 1.21·41-s − 6.83·43-s + 8.43·47-s + 3.33·49-s + 12.1·53-s − 4.38·55-s + 8.76·59-s − 11.4·61-s − 4.38·65-s + 8.81·67-s − 4.69·71-s + 13.9·73-s + 3.21·77-s − 13.6·79-s + 8.76·83-s + ⋯
L(s)  = 1  − 1.96·5-s + 1.21·7-s + 0.301·11-s + 0.277·13-s + 0.452·17-s − 1.41·19-s − 0.726·23-s + 2.84·25-s − 0.525·29-s − 0.194·31-s − 2.38·35-s − 0.443·37-s + 0.189·41-s − 1.04·43-s + 1.22·47-s + 0.476·49-s + 1.67·53-s − 0.591·55-s + 1.14·59-s − 1.46·61-s − 0.543·65-s + 1.07·67-s − 0.557·71-s + 1.62·73-s + 0.366·77-s − 1.53·79-s + 0.962·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5148\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(41.1069\)
Root analytic conductor: \(6.41147\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5148,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + 4.38T + 5T^{2} \)
7 \( 1 - 3.21T + 7T^{2} \)
17 \( 1 - 1.86T + 17T^{2} \)
19 \( 1 + 6.18T + 19T^{2} \)
23 \( 1 + 3.48T + 23T^{2} \)
29 \( 1 + 2.83T + 29T^{2} \)
31 \( 1 + 1.08T + 31T^{2} \)
37 \( 1 + 2.69T + 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 + 6.83T + 43T^{2} \)
47 \( 1 - 8.43T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 - 8.76T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 8.81T + 67T^{2} \)
71 \( 1 + 4.69T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 8.76T + 83T^{2} \)
89 \( 1 + 18.4T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−7.908346630945058605803019397820, −7.33210201838510571096815411661, −6.63391564033739837977586452982, −5.54550477560236401621035949951, −4.71551204658449262275336724623, −4.04302514029693256308359429099, −3.64973713122340751002098380469, −2.38249598265627917381333324838, −1.22561188478423358051213009590, 0, 1.22561188478423358051213009590, 2.38249598265627917381333324838, 3.64973713122340751002098380469, 4.04302514029693256308359429099, 4.71551204658449262275336724623, 5.54550477560236401621035949951, 6.63391564033739837977586452982, 7.33210201838510571096815411661, 7.908346630945058605803019397820

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