Properties

Label 2-25200-1.1-c1-0-106
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 − 4·13-s + 6·17-s − 6·19-s + 8·23-s + 2·29-s − 10·31-s − 2·37-s − 10·41-s − 4·43-s + 8·47-s + 49-s + 4·53-s − 8·59-s + 6·61-s + 12·67-s − 6·71-s + 12·73-s − 2·77-s + 8·79-s + 4·83-s + 10·89-s + 4·91-s − 8·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.603·11-s − 1.10·13-s + 1.45·17-s − 1.37·19-s + 1.66·23-s + 0.371·29-s − 1.79·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s − 1.04·59-s + 0.768·61-s + 1.46·67-s − 0.712·71-s + 1.40·73-s − 0.227·77-s + 0.900·79-s + 0.439·83-s + 1.05·89-s + 0.419·91-s − 0.812·97-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: $-1$
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−15.40449273717015, −15.03389731609236, −14.65923442467843, −14.12208146235760, −13.46370129084106, −12.81122959572959, −12.40805932923096, −12.02749351637510, −11.28594725111076, −10.64571411562821, −10.24691225883780, −9.516313577771579, −9.148947316041449, −8.495741466934525, −7.850454613868564, −7.076612438577563, −6.846777189285347, −6.080412304202392, −5.193978630149594, −5.025757839090358, −3.928242706718422, −3.499509586250852, −2.696466296394976, −1.938382035049136, −1.035122734538915, 0, 1.035122734538915, 1.938382035049136, 2.696466296394976, 3.499509586250852, 3.928242706718422, 5.025757839090358, 5.193978630149594, 6.080412304202392, 6.846777189285347, 7.076612438577563, 7.850454613868564, 8.495741466934525, 9.148947316041449, 9.516313577771579, 10.24691225883780, 10.64571411562821, 11.28594725111076, 12.02749351637510, 12.40805932923096, 12.81122959572959, 13.46370129084106, 14.12208146235760, 14.65923442467843, 15.03389731609236, 15.40449273717015

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