Properties

Label 2-5292-1.1-c1-0-33
Degree $2$
Conductor $5292$
Sign $-1$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  − 2.23·5-s − 0.540·11-s − 2.62·13-s + 1.47·17-s + 4.03·19-s + 3.16·23-s + 0.540·29-s − 2.62·31-s + 4.70·37-s − 2.23·41-s + 8.70·43-s − 4.52·47-s − 11.6·53-s + 1.20·55-s − 11.9·59-s + 9.69·61-s + 5.86·65-s + 3.70·67-s + 13.7·71-s − 1.41·73-s + 5·79-s − 3.76·83-s − 3.29·85-s − 6.76·89-s − 9.02·95-s − 9.69·97-s − 9.70·101-s + ⋯
L(s)  = 1  − 0.999·5-s − 0.162·11-s − 0.727·13-s + 0.357·17-s + 0.925·19-s + 0.659·23-s + 0.100·29-s − 0.470·31-s + 0.774·37-s − 0.349·41-s + 1.32·43-s − 0.660·47-s − 1.59·53-s + 0.162·55-s − 1.55·59-s + 1.24·61-s + 0.727·65-s + 0.453·67-s + 1.62·71-s − 0.165·73-s + 0.562·79-s − 0.413·83-s − 0.357·85-s − 0.716·89-s − 0.925·95-s − 0.984·97-s − 0.966·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.23T + 5T^{2} \)
11 \( 1 + 0.540T + 11T^{2} \)
13 \( 1 + 2.62T + 13T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 - 4.03T + 19T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 - 0.540T + 29T^{2} \)
31 \( 1 + 2.62T + 31T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 + 2.23T + 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + 4.52T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 9.69T + 61T^{2} \)
67 \( 1 - 3.70T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + 3.76T + 83T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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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

−7.81822101582926987005591401850, −7.28330860631428663984240674811, −6.53416551456849379116092558442, −5.52643915688983388571527849567, −4.90377420764899516353922556178, −4.08263313341875669864204960796, −3.31525879998111085804430572265, −2.53141630294585718966260267651, −1.21179326945227900861678351665, 0, 1.21179326945227900861678351665, 2.53141630294585718966260267651, 3.31525879998111085804430572265, 4.08263313341875669864204960796, 4.90377420764899516353922556178, 5.52643915688983388571527849567, 6.53416551456849379116092558442, 7.28330860631428663984240674811, 7.81822101582926987005591401850

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