Properties

Label 2-6552-1.1-c1-0-30
Degree $2$
Conductor $6552$
Sign $1$
Analytic cond. $52.3179$
Root an. cond. $7.23311$
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  + 3·5-s + 7-s − 6·11-s − 13-s + 8·17-s − 19-s − 23-s + 4·25-s + 5·29-s + 3·31-s + 3·35-s + 12·37-s − 10·41-s − 11·43-s + 3·47-s + 49-s + 53-s − 18·55-s + 8·59-s + 2·61-s − 3·65-s + 2·67-s − 8·71-s + 11·73-s − 6·77-s + 13·79-s + 7·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 1.94·17-s − 0.229·19-s − 0.208·23-s + 4/5·25-s + 0.928·29-s + 0.538·31-s + 0.507·35-s + 1.97·37-s − 1.56·41-s − 1.67·43-s + 0.437·47-s + 1/7·49-s + 0.137·53-s − 2.42·55-s + 1.04·59-s + 0.256·61-s − 0.372·65-s + 0.244·67-s − 0.949·71-s + 1.28·73-s − 0.683·77-s + 1.46·79-s + 0.768·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6552\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(52.3179\)
Root analytic conductor: \(7.23311\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6552,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.632398208\)
\(L(\frac12)\) \(\approx\) \(2.632398208\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.076338535830487059327373832421, −7.40932557585347156543741399302, −6.46212285818300468296740723283, −5.75996429551651431345261301215, −5.22681463243730871841155306889, −4.73229481346635177444876521333, −3.37811428815993667385712448598, −2.63047241791502753917662926061, −1.94786199622203952249066764095, −0.844940816180295573424825489467, 0.844940816180295573424825489467, 1.94786199622203952249066764095, 2.63047241791502753917662926061, 3.37811428815993667385712448598, 4.73229481346635177444876521333, 5.22681463243730871841155306889, 5.75996429551651431345261301215, 6.46212285818300468296740723283, 7.40932557585347156543741399302, 8.076338535830487059327373832421

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