Properties

Label 2-58800-1.1-c1-0-106
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 − 6·11-s + 5·13-s − 6·17-s + 5·19-s − 6·23-s − 27-s − 6·29-s − 31-s + 6·33-s − 2·37-s − 5·39-s − 43-s − 6·47-s + 6·51-s + 12·53-s − 5·57-s − 6·59-s + 13·61-s + 11·67-s + 6·69-s + 2·73-s − 8·79-s + 81-s + 6·83-s + 6·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.38·13-s − 1.45·17-s + 1.14·19-s − 1.25·23-s − 0.192·27-s − 1.11·29-s − 0.179·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s − 0.152·43-s − 0.875·47-s + 0.840·51-s + 1.64·53-s − 0.662·57-s − 0.781·59-s + 1.66·61-s + 1.34·67-s + 0.722·69-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.658·83-s + 0.643·87-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.56408312303212, −13.94213956873909, −13.28472936323399, −13.25937022594090, −12.72837353274949, −11.89063581939062, −11.53490097197637, −10.91516978279511, −10.72000961802173, −10.01415924908824, −9.613227637066306, −8.820025413364233, −8.344538929216669, −7.851416658571146, −7.256593655407632, −6.691434088566023, −6.053573462854549, −5.479320955394333, −5.214549328857454, −4.379849736484535, −3.801510771022759, −3.178753328126813, −2.291042289879081, −1.820468025319226, −0.7612954584705766, 0, 0.7612954584705766, 1.820468025319226, 2.291042289879081, 3.178753328126813, 3.801510771022759, 4.379849736484535, 5.214549328857454, 5.479320955394333, 6.053573462854549, 6.691434088566023, 7.256593655407632, 7.851416658571146, 8.344538929216669, 8.820025413364233, 9.613227637066306, 10.01415924908824, 10.72000961802173, 10.91516978279511, 11.53490097197637, 11.89063581939062, 12.72837353274949, 13.25937022594090, 13.28472936323399, 13.94213956873909, 14.56408312303212

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