Properties

Label 2-58800-1.1-c1-0-138
Degree $2$
Conductor $58800$
Sign $-1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
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  − 3-s + 9-s − 2·13-s + 6·17-s − 2·19-s − 27-s − 4·31-s − 6·37-s + 2·39-s + 2·41-s − 10·43-s − 2·47-s − 6·51-s + 6·53-s + 2·57-s + 12·59-s + 10·61-s − 10·67-s − 6·71-s − 10·73-s + 4·79-s + 81-s + 4·83-s − 6·89-s + 4·93-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.458·19-s − 0.192·27-s − 0.718·31-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 1.52·43-s − 0.291·47-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.28·61-s − 1.22·67-s − 0.712·71-s − 1.17·73-s + 0.450·79-s + 1/9·81-s + 0.439·83-s − 0.635·89-s + 0.414·93-s − 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.54165800361942, −14.24911358093577, −13.46689864826124, −12.98778061556104, −12.56823902174966, −11.90382601730465, −11.72747825421294, −11.07135093245331, −10.38051493375541, −10.04546260951768, −9.703734163884049, −8.740474398217290, −8.539667475872337, −7.646184584166854, −7.283013166450229, −6.774627199305558, −6.020317468548420, −5.584097539755330, −5.042501872521200, −4.510630301166531, −3.662465609501234, −3.281927792752530, −2.336926925599582, −1.683687970322015, −0.8695334885819742, 0, 0.8695334885819742, 1.683687970322015, 2.336926925599582, 3.281927792752530, 3.662465609501234, 4.510630301166531, 5.042501872521200, 5.584097539755330, 6.020317468548420, 6.774627199305558, 7.283013166450229, 7.646184584166854, 8.539667475872337, 8.740474398217290, 9.703734163884049, 10.04546260951768, 10.38051493375541, 11.07135093245331, 11.72747825421294, 11.90382601730465, 12.56823902174966, 12.98778061556104, 13.46689864826124, 14.24911358093577, 14.54165800361942

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