Properties

Degree $2$
Conductor $193600$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·7-s + 6·9-s − 4·13-s + 4·19-s + 9·21-s + 8·23-s − 9·27-s + 6·29-s − 2·31-s − 8·37-s + 12·39-s + 5·41-s − 5·43-s + 3·47-s + 2·49-s + 4·53-s − 12·57-s + 2·59-s − 11·61-s − 18·63-s − 13·67-s − 24·69-s + 2·71-s − 8·73-s − 10·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.13·7-s + 2·9-s − 1.10·13-s + 0.917·19-s + 1.96·21-s + 1.66·23-s − 1.73·27-s + 1.11·29-s − 0.359·31-s − 1.31·37-s + 1.92·39-s + 0.780·41-s − 0.762·43-s + 0.437·47-s + 2/7·49-s + 0.549·53-s − 1.58·57-s + 0.260·59-s − 1.40·61-s − 2.26·63-s − 1.58·67-s − 2.88·69-s + 0.237·71-s − 0.936·73-s − 1.12·79-s + 81-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: \(0\)
Selberg data: \((2,\ 193600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2307247799\)
\(L(\frac12)\) \(\approx\) \(0.2307247799\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.79914242950763, −12.56544916019467, −12.14364281607609, −11.74366722771165, −11.31010058955473, −10.69348946997216, −10.26630901870260, −10.07272512056740, −9.298880071464851, −9.148187534084676, −8.340128270794270, −7.352427616682824, −7.304028258676451, −6.752917596245582, −6.347711051707878, −5.729857522305639, −5.351654536767543, −4.836419213405828, −4.475364715286892, −3.675570183997333, −3.008361548020318, −2.627011708451327, −1.499798222300144, −1.029228200212497, −0.1800769474106760, 0.1800769474106760, 1.029228200212497, 1.499798222300144, 2.627011708451327, 3.008361548020318, 3.675570183997333, 4.475364715286892, 4.836419213405828, 5.351654536767543, 5.729857522305639, 6.347711051707878, 6.752917596245582, 7.304028258676451, 7.352427616682824, 8.340128270794270, 9.148187534084676, 9.298880071464851, 10.07272512056740, 10.26630901870260, 10.69348946997216, 11.31010058955473, 11.74366722771165, 12.14364281607609, 12.56544916019467, 12.79914242950763

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