Properties

Degree $2$
Conductor $33600$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

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: \(0\)
Selberg data: \((2,\ 33600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.776565642\)
\(L(\frac12)\) \(\approx\) \(3.776565642\)
\(L(\frac{3}{2})\) 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

−14.81172241255997, −14.44914975481789, −14.02123918107304, −13.47758956855566, −12.94059716248137, −12.36970163327113, −11.95407801768921, −11.31687588513157, −10.61402195149556, −10.11874867337876, −9.686985476544381, −8.964237584528163, −8.691971790730617, −7.779192670589494, −7.527978198821159, −6.910373755618202, −6.061722563158145, −5.649002142224710, −5.000758071910208, −3.999391137522663, −3.745433923453942, −2.862451690219310, −2.471506616834313, −1.288380896389549, −0.8042775637284579, 0.8042775637284579, 1.288380896389549, 2.471506616834313, 2.862451690219310, 3.745433923453942, 3.999391137522663, 5.000758071910208, 5.649002142224710, 6.061722563158145, 6.910373755618202, 7.527978198821159, 7.779192670589494, 8.691971790730617, 8.964237584528163, 9.686985476544381, 10.11874867337876, 10.61402195149556, 11.31687588513157, 11.95407801768921, 12.36970163327113, 12.94059716248137, 13.47758956855566, 14.02123918107304, 14.44914975481789, 14.81172241255997

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