Properties

Degree $2$
Conductor $193600$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s + 13-s + 5·17-s + 6·19-s + 4·21-s + 2·23-s + 4·27-s − 9·29-s + 2·31-s − 3·37-s − 2·39-s − 5·41-s + 2·47-s − 3·49-s − 10·51-s + 9·53-s − 12·57-s + 8·59-s − 6·61-s − 2·63-s − 2·67-s − 4·69-s − 12·71-s + 2·73-s + 10·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 1.21·17-s + 1.37·19-s + 0.872·21-s + 0.417·23-s + 0.769·27-s − 1.67·29-s + 0.359·31-s − 0.493·37-s − 0.320·39-s − 0.780·41-s + 0.291·47-s − 3/7·49-s − 1.40·51-s + 1.23·53-s − 1.58·57-s + 1.04·59-s − 0.768·61-s − 0.251·63-s − 0.244·67-s − 0.481·69-s − 1.42·71-s + 0.234·73-s + 1.12·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{193600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 193600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.21253159753780, −12.83182095126134, −12.24583960597642, −11.89472552831436, −11.49546471095755, −11.14098450244559, −10.39464398334060, −10.17194007399190, −9.689424782722561, −9.110274515132107, −8.703872843481090, −7.968255972336118, −7.401358223862875, −7.096228453067164, −6.482139977014445, −5.930460528086557, −5.569897867787477, −5.227696085721291, −4.636677009136552, −3.836015922119354, −3.315219029753506, −3.003341863327182, −2.050620108880683, −1.261434748996488, −0.7438548984505386, 0, 0.7438548984505386, 1.261434748996488, 2.050620108880683, 3.003341863327182, 3.315219029753506, 3.836015922119354, 4.636677009136552, 5.227696085721291, 5.569897867787477, 5.930460528086557, 6.482139977014445, 7.096228453067164, 7.401358223862875, 7.968255972336118, 8.703872843481090, 9.110274515132107, 9.689424782722561, 10.17194007399190, 10.39464398334060, 11.14098450244559, 11.49546471095755, 11.89472552831436, 12.24583960597642, 12.83182095126134, 13.21253159753780

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