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Properties

Label 2-8664-1.1-c1-0-170
Degree $2$
Conductor $8664$
Sign $1$
Analytic cond. $69.1823$
Root an. cond. $8.31759$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 5·7-s + 9-s − 4·11-s − 5·13-s − 2·15-s − 5·21-s − 6·23-s − 25-s + 27-s − 8·29-s + 31-s − 4·33-s + 10·35-s − 7·37-s − 5·39-s − 11·43-s − 2·45-s + 10·47-s + 18·49-s − 6·53-s + 8·55-s − 8·59-s − 61-s − 5·63-s + 10·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.88·7-s + 1/3·9-s − 1.20·11-s − 1.38·13-s − 0.516·15-s − 1.09·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.48·29-s + 0.179·31-s − 0.696·33-s + 1.69·35-s − 1.15·37-s − 0.800·39-s − 1.67·43-s − 0.298·45-s + 1.45·47-s + 18/7·49-s − 0.824·53-s + 1.07·55-s − 1.04·59-s − 0.128·61-s − 0.629·63-s + 1.24·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8664\)    =    \(2^{3} \cdot 3 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(69.1823\)
Root analytic conductor: \(8.31759\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 8664,\ (\ :1/2),\ 1)\)

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 \( 1 \)
3 \( 1 - T \)
19 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.33869438743301300406859435816, −6.52191544606770970357436859699, −5.73445014165525802841585152884, −4.94373805834911063585061854116, −3.99280860593233180838870043353, −3.44196122614469365488044467428, −2.78561902359244845453670190594, −2.03984449878263612023368049843, 0, 0, 2.03984449878263612023368049843, 2.78561902359244845453670190594, 3.44196122614469365488044467428, 3.99280860593233180838870043353, 4.94373805834911063585061854116, 5.73445014165525802841585152884, 6.52191544606770970357436859699, 7.33869438743301300406859435816

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