Properties

Label 2-8624-1.1-c1-0-76
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 3·9-s + 11-s − 2·13-s + 4·17-s − 6·19-s − 4·23-s + 11·25-s − 2·29-s − 2·31-s + 10·37-s − 4·41-s + 8·43-s − 12·45-s + 2·47-s + 6·53-s + 4·55-s − 12·59-s + 14·61-s − 8·65-s + 12·67-s + 8·71-s − 4·73-s + 9·81-s − 6·83-s + 16·85-s + 6·89-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-s + 0.301·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 0.834·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 1.64·37-s − 0.624·41-s + 1.21·43-s − 1.78·45-s + 0.291·47-s + 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.79·61-s − 0.992·65-s + 1.46·67-s + 0.949·71-s − 0.468·73-s + 81-s − 0.658·83-s + 1.73·85-s + 0.635·89-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8624,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.677767221\)
\(L(\frac12)\) \(\approx\) \(2.677767221\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.84243688128141633065550543669, −6.92743859216994593885061344591, −6.13084714923078385076923190925, −5.85808430856797751118865780237, −5.22279012395712767950269184635, −4.34539946912301064945419937597, −3.32636164825504490335628616164, −2.35792223982950421513090216645, −2.04253952139450276371464204879, −0.78385397243241047554565051843, 0.78385397243241047554565051843, 2.04253952139450276371464204879, 2.35792223982950421513090216645, 3.32636164825504490335628616164, 4.34539946912301064945419937597, 5.22279012395712767950269184635, 5.85808430856797751118865780237, 6.13084714923078385076923190925, 6.92743859216994593885061344591, 7.84243688128141633065550543669

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