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Properties

Label 2-5292-1.1-c1-0-17
Degree $2$
Conductor $5292$
Sign $1$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·13-s + 8·19-s − 5·25-s − 7·31-s + 11·37-s + 5·43-s − 13·61-s + 5·67-s − 10·73-s + 17·79-s + 5·97-s − 13·103-s + 17·109-s + ⋯
L(s)  = 1  + 1.38·13-s + 1.83·19-s − 25-s − 1.25·31-s + 1.80·37-s + 0.762·43-s − 1.66·61-s + 0.610·67-s − 1.17·73-s + 1.91·79-s + 0.507·97-s − 1.28·103-s + 1.62·109-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.199005564\)
\(L(\frac12)\) \(\approx\) \(2.199005564\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−7.946240313775152625296996238812, −7.69803181621610360495425070984, −6.74420295625752950740620167182, −5.89799281763541579861161429354, −5.50370149225684586936100752165, −4.42425178073269823498573562975, −3.66787536854896765277410826765, −2.96894684880080108642154400584, −1.77894635329285426187308770868, −0.839698587098722534959114384229, 0.839698587098722534959114384229, 1.77894635329285426187308770868, 2.96894684880080108642154400584, 3.66787536854896765277410826765, 4.42425178073269823498573562975, 5.50370149225684586936100752165, 5.89799281763541579861161429354, 6.74420295625752950740620167182, 7.69803181621610360495425070984, 7.946240313775152625296996238812

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