Properties

Degree $2$
Conductor $25200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 5·11-s − 6·13-s − 4·17-s + 6·19-s + 3·23-s + 3·29-s − 2·31-s + 7·37-s + 4·41-s + 7·43-s − 2·47-s + 49-s + 10·53-s − 14·59-s + 4·61-s − 3·67-s − 13·71-s + 16·73-s + 5·77-s − 79-s + 10·83-s − 10·89-s + 6·91-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.50·11-s − 1.66·13-s − 0.970·17-s + 1.37·19-s + 0.625·23-s + 0.557·29-s − 0.359·31-s + 1.15·37-s + 0.624·41-s + 1.06·43-s − 0.291·47-s + 1/7·49-s + 1.37·53-s − 1.82·59-s + 0.512·61-s − 0.366·67-s − 1.54·71-s + 1.87·73-s + 0.569·77-s − 0.112·79-s + 1.09·83-s − 1.05·89-s + 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{25200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
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

−15.66385713978948, −15.08244182823572, −14.69745633255880, −13.89063506921034, −13.49621036315883, −12.93620411130578, −12.41536249689245, −12.00153694323324, −11.18028835644929, −10.77158730425956, −10.14603509270913, −9.541902064011089, −9.267616425891946, −8.384619320003783, −7.695973862139053, −7.361872983443768, −6.821696589331237, −5.910314838402926, −5.376461765606819, −4.791985613722574, −4.283721012263535, −3.155866568048451, −2.686584872385702, −2.180532574409666, −0.8741527135138022, 0, 0.8741527135138022, 2.180532574409666, 2.686584872385702, 3.155866568048451, 4.283721012263535, 4.791985613722574, 5.376461765606819, 5.910314838402926, 6.821696589331237, 7.361872983443768, 7.695973862139053, 8.384619320003783, 9.267616425891946, 9.541902064011089, 10.14603509270913, 10.77158730425956, 11.18028835644929, 12.00153694323324, 12.41536249689245, 12.93620411130578, 13.49621036315883, 13.89063506921034, 14.69745633255880, 15.08244182823572, 15.66385713978948

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