Properties

Label 2-25200-1.1-c1-0-135
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 + 2·11-s + 6·13-s − 4·17-s + 4·19-s − 2·23-s + 2·29-s − 2·37-s − 4·43-s − 12·47-s + 49-s − 6·53-s − 8·59-s + 6·61-s − 8·67-s + 14·71-s + 2·73-s − 2·77-s − 12·79-s + 4·83-s − 6·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.603·11-s + 1.66·13-s − 0.970·17-s + 0.917·19-s − 0.417·23-s + 0.371·29-s − 0.328·37-s − 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.04·59-s + 0.768·61-s − 0.977·67-s + 1.66·71-s + 0.234·73-s − 0.227·77-s − 1.35·79-s + 0.439·83-s − 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−15.88541683638899, −15.13656378010121, −14.54772231600569, −13.85941037768347, −13.53640869231775, −13.09058458349636, −12.35412752101343, −11.85452918000077, −11.14755537879427, −10.97610983071725, −10.12455145454971, −9.541820128485428, −9.098122557296207, −8.372135528135169, −8.086222864958641, −7.111316938704445, −6.550926264288718, −6.212671926806157, −5.465471114911217, −4.730347664536197, −3.983110183524777, −3.473792039984687, −2.816739764707330, −1.741022622465732, −1.171921253410920, 0, 1.171921253410920, 1.741022622465732, 2.816739764707330, 3.473792039984687, 3.983110183524777, 4.730347664536197, 5.465471114911217, 6.212671926806157, 6.550926264288718, 7.111316938704445, 8.086222864958641, 8.372135528135169, 9.098122557296207, 9.541820128485428, 10.12455145454971, 10.97610983071725, 11.14755537879427, 11.85452918000077, 12.35412752101343, 13.09058458349636, 13.53640869231775, 13.85941037768347, 14.54772231600569, 15.13656378010121, 15.88541683638899

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