Properties

Degree $2$
Conductor $33600$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s + 2·13-s + 6·17-s − 4·19-s − 21-s − 27-s + 6·29-s + 4·31-s + 2·37-s − 2·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s − 6·51-s + 6·53-s + 4·57-s − 12·59-s − 2·61-s + 63-s − 8·67-s − 14·73-s + 16·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.125·63-s − 0.977·67-s − 1.63·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{33600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−15.19435013075614, −14.74079063539729, −14.32255792657175, −13.52760976331861, −13.27374096812870, −12.46632253832686, −12.04194016263953, −11.69736931167879, −10.90345543182341, −10.58909730566052, −10.00801397089079, −9.476452783624830, −8.739298554019674, −8.031433596981929, −7.898556427725120, −6.901058693766526, −6.458002341338438, −5.872004593082327, −5.306007953760068, −4.614096812217202, −4.146399435665722, −3.273175048766160, −2.677493388614919, −1.578154585296770, −1.114408695829257, 0, 1.114408695829257, 1.578154585296770, 2.677493388614919, 3.273175048766160, 4.146399435665722, 4.614096812217202, 5.306007953760068, 5.872004593082327, 6.458002341338438, 6.901058693766526, 7.898556427725120, 8.031433596981929, 8.739298554019674, 9.476452783624830, 10.00801397089079, 10.58909730566052, 10.90345543182341, 11.69736931167879, 12.04194016263953, 12.46632253832686, 13.27374096812870, 13.52760976331861, 14.32255792657175, 14.74079063539729, 15.19435013075614

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