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 − 4·7-s + 9-s − 4·13-s + 4·17-s − 4·19-s + 8·21-s − 6·23-s + 4·27-s − 10·29-s + 4·31-s + 2·37-s + 8·39-s + 10·41-s + 8·43-s + 6·47-s + 9·49-s − 8·51-s + 2·53-s + 8·57-s + 4·59-s − 10·61-s − 4·63-s − 2·67-s + 12·69-s + 12·73-s − 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s + 0.970·17-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s − 1.85·29-s + 0.718·31-s + 0.328·37-s + 1.28·39-s + 1.56·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 1.12·51-s + 0.274·53-s + 1.05·57-s + 0.520·59-s − 1.28·61-s − 0.503·63-s − 0.244·67-s + 1.44·69-s + 1.40·73-s − 0.900·79-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.09095652375622, −12.60208014488503, −12.46833740837042, −12.04506804883130, −11.50745524193179, −10.92359053065024, −10.56756648182477, −9.976040837365664, −9.724950707038819, −9.233657142545142, −8.713202499623408, −7.817049528892021, −7.582541262433336, −6.973509652983598, −6.429885611878373, −5.958124144424759, −5.730692312957941, −5.216175166507378, −4.383041852361826, −4.043487830908272, −3.426374819739226, −2.574897614534365, −2.402125360799003, −1.303301648932222, −0.4970221426565379, 0, 0.4970221426565379, 1.303301648932222, 2.402125360799003, 2.574897614534365, 3.426374819739226, 4.043487830908272, 4.383041852361826, 5.216175166507378, 5.730692312957941, 5.958124144424759, 6.429885611878373, 6.973509652983598, 7.582541262433336, 7.817049528892021, 8.713202499623408, 9.233657142545142, 9.724950707038819, 9.976040837365664, 10.56756648182477, 10.92359053065024, 11.50745524193179, 12.04506804883130, 12.46833740837042, 12.60208014488503, 13.09095652375622

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