Properties

Label 2-25200-1.1-c1-0-110
Degree $2$
Conductor $25200$
Sign $-1$
Analytic cond. $201.223$
Root an. cond. $14.1853$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 11-s − 2·13-s + 4·17-s + 2·19-s − 5·23-s − 29-s + 2·31-s − 3·37-s − 12·41-s + 11·43-s − 2·47-s + 49-s + 6·53-s − 10·59-s + 4·61-s + 67-s − 3·71-s − 77-s + 9·79-s + 2·83-s + 6·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.301·11-s − 0.554·13-s + 0.970·17-s + 0.458·19-s − 1.04·23-s − 0.185·29-s + 0.359·31-s − 0.493·37-s − 1.87·41-s + 1.67·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s − 1.30·59-s + 0.512·61-s + 0.122·67-s − 0.356·71-s − 0.113·77-s + 1.01·79-s + 0.219·83-s + 0.635·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: $-1$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
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

−15.62909409442327, −15.11030645864763, −14.52229293506265, −13.95613631789118, −13.65876263743255, −12.87741459828315, −12.24085494952945, −12.02627466773539, −11.41513653017299, −10.60963920373455, −10.11973416381198, −9.691993342401546, −9.103891276860232, −8.449222188013173, −7.776309234254699, −7.348959257845815, −6.632928621775109, −6.050212368817270, −5.416353062300799, −4.839576204713527, −3.980860459549969, −3.461858747286737, −2.723351202830636, −1.917846329902571, −1.044101483706880, 0, 1.044101483706880, 1.917846329902571, 2.723351202830636, 3.461858747286737, 3.980860459549969, 4.839576204713527, 5.416353062300799, 6.050212368817270, 6.632928621775109, 7.348959257845815, 7.776309234254699, 8.449222188013173, 9.103891276860232, 9.691993342401546, 10.11973416381198, 10.60963920373455, 11.41513653017299, 12.02627466773539, 12.24085494952945, 12.87741459828315, 13.65876263743255, 13.95613631789118, 14.52229293506265, 15.11030645864763, 15.62909409442327

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