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Properties

Label 2-8664-1.1-c1-0-59
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 $1$

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: \(1\)
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

−7.26862120002587939985316867098, −6.73157571455056078268250945492, −5.96922401937799811451688747390, −5.65950625494465729059763152632, −4.46179479774421579353123969079, −3.82771111588679343959885834948, −3.22715035707737567492334020229, −2.38382953353414703435598484617, −0.825526731535079557530284340505, 0, 0.825526731535079557530284340505, 2.38382953353414703435598484617, 3.22715035707737567492334020229, 3.82771111588679343959885834948, 4.46179479774421579353123969079, 5.65950625494465729059763152632, 5.96922401937799811451688747390, 6.73157571455056078268250945492, 7.26862120002587939985316867098

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