Properties

Label 2-8624-1.1-c1-0-113
Degree $2$
Conductor $8624$
Sign $-1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s − 11-s − 4·13-s + 15-s + 2·17-s + 23-s − 4·25-s + 5·27-s + 7·31-s + 33-s + 3·37-s + 4·39-s + 8·41-s + 6·43-s + 2·45-s + 8·47-s − 2·51-s − 6·53-s + 55-s + 5·59-s − 12·61-s + 4·65-s + 7·67-s − 69-s + 3·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s + 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.25·31-s + 0.174·33-s + 0.493·37-s + 0.640·39-s + 1.24·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.134·55-s + 0.650·59-s − 1.53·61-s + 0.496·65-s + 0.855·67-s − 0.120·69-s + 0.356·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8624\)    =    \(2^{4} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(68.8629\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8624,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.59753326202792059542705479631, −6.69785642750182140490053057916, −5.96253635220307759773371824019, −5.41592840817145656472425824771, −4.67360648409355241573689732264, −4.01118939616306336398601307633, −2.92516350065322057627024861621, −2.40668113522951484990555404309, −0.979033992832471368219893964268, 0, 0.979033992832471368219893964268, 2.40668113522951484990555404309, 2.92516350065322057627024861621, 4.01118939616306336398601307633, 4.67360648409355241573689732264, 5.41592840817145656472425824771, 5.96253635220307759773371824019, 6.69785642750182140490053057916, 7.59753326202792059542705479631

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