Properties

Label 2-58800-1.1-c1-0-179
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 − 4·19-s − 27-s − 6·29-s + 8·31-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 6·51-s + 6·53-s + 4·57-s + 10·61-s − 4·67-s + 2·73-s − 8·79-s + 81-s − 12·83-s + 6·87-s − 18·89-s − 8·93-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.488·67-s + 0.234·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-s − 1.90·89-s − 0.829·93-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.69803123199372, −14.02742943079618, −13.57276852114818, −12.98807311755048, −12.51666437770645, −12.11107739309543, −11.43773747850965, −11.16415952578223, −10.46934625330928, −10.02273973860259, −9.645174474422924, −8.847739417094762, −8.343594077989274, −7.876658289370419, −7.167884073830039, −6.733197711629061, −6.001912310661009, −5.658834570619837, −5.108876070241209, −4.274512109042676, −3.914465746479856, −3.131040567196598, −2.449351378854893, −1.542042644132964, −0.9693927657663917, 0, 0.9693927657663917, 1.542042644132964, 2.449351378854893, 3.131040567196598, 3.914465746479856, 4.274512109042676, 5.108876070241209, 5.658834570619837, 6.001912310661009, 6.733197711629061, 7.167884073830039, 7.876658289370419, 8.343594077989274, 8.847739417094762, 9.645174474422924, 10.02273973860259, 10.46934625330928, 11.16415952578223, 11.43773747850965, 12.11107739309543, 12.51666437770645, 12.98807311755048, 13.57276852114818, 14.02742943079618, 14.69803123199372

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