Properties

Label 2-8624-1.1-c1-0-133
Degree $2$
Conductor $8624$
Sign $-1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
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  − 2.44·3-s + 1.41·5-s + 2.99·9-s − 11-s + 1.03·13-s − 3.46·15-s + 1.03·17-s − 0.535·23-s − 2.99·25-s + 3.46·29-s − 5.27·31-s + 2.44·33-s + 2·37-s − 2.53·39-s + 6.69·41-s − 5.46·43-s + 4.24·45-s + 0.378·47-s − 2.53·51-s − 6.92·53-s − 1.41·55-s − 8.10·59-s + 3.86·61-s + 1.46·65-s − 4.53·67-s + 1.31·69-s − 6.39·71-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.632·5-s + 0.999·9-s − 0.301·11-s + 0.287·13-s − 0.894·15-s + 0.251·17-s − 0.111·23-s − 0.599·25-s + 0.643·29-s − 0.947·31-s + 0.426·33-s + 0.328·37-s − 0.406·39-s + 1.04·41-s − 0.833·43-s + 0.632·45-s + 0.0552·47-s − 0.355·51-s − 0.951·53-s − 0.190·55-s − 1.05·59-s + 0.494·61-s + 0.181·65-s − 0.554·67-s + 0.158·69-s − 0.758·71-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 2.44T + 3T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
13 \( 1 - 1.03T + 13T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 0.535T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 5.27T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 6.69T + 41T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 - 0.378T + 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + 8.10T + 59T^{2} \)
61 \( 1 - 3.86T + 61T^{2} \)
67 \( 1 + 4.53T + 67T^{2} \)
71 \( 1 + 6.39T + 71T^{2} \)
73 \( 1 - 9.52T + 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 - 2.07T + 83T^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 - 0.656T + 97T^{2} \)
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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

−7.31513097620195005393581530781, −6.42697724467303570728744638468, −6.08423988944490430120387085914, −5.42657063920125394155661875507, −4.88357449934237698704562413057, −4.06306825365886045405485368723, −3.06243826288617860731427623447, −2.01736309412868891379033210669, −1.09991191362984260549444044840, 0, 1.09991191362984260549444044840, 2.01736309412868891379033210669, 3.06243826288617860731427623447, 4.06306825365886045405485368723, 4.88357449934237698704562413057, 5.42657063920125394155661875507, 6.08423988944490430120387085914, 6.42697724467303570728744638468, 7.31513097620195005393581530781

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