Properties

Label 2-9600-1.1-c1-0-86
Degree $2$
Conductor $9600$
Sign $-1$
Analytic cond. $76.6563$
Root an. cond. $8.75536$
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 + 9-s − 2·11-s − 4·13-s − 27-s + 2·29-s + 10·31-s + 2·33-s − 4·37-s + 4·39-s + 6·41-s + 4·43-s − 7·49-s − 2·53-s − 6·59-s + 6·61-s + 4·67-s + 8·71-s − 14·73-s + 2·79-s + 81-s + 4·83-s − 2·87-s + 14·89-s − 10·93-s + 6·97-s − 2·99-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.192·27-s + 0.371·29-s + 1.79·31-s + 0.348·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 49-s − 0.274·53-s − 0.781·59-s + 0.768·61-s + 0.488·67-s + 0.949·71-s − 1.63·73-s + 0.225·79-s + 1/9·81-s + 0.439·83-s − 0.214·87-s + 1.48·89-s − 1.03·93-s + 0.609·97-s − 0.201·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9600\)    =    \(2^{7} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(76.6563\)
Root analytic conductor: \(8.75536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.42137206045731387956571558615, −6.55143673846532172969865055424, −6.08196247337318922483529380318, −5.08202663433445960270772560552, −4.83478249552697908292050480823, −3.93632783998710463972534562493, −2.89500726761865448274516487337, −2.26435217617127305427134080862, −1.07146533574569719267393540090, 0, 1.07146533574569719267393540090, 2.26435217617127305427134080862, 2.89500726761865448274516487337, 3.93632783998710463972534562493, 4.83478249552697908292050480823, 5.08202663433445960270772560552, 6.08196247337318922483529380318, 6.55143673846532172969865055424, 7.42137206045731387956571558615

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