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Properties

Label 2-58800-1.1-c1-0-66
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 5·11-s − 2·13-s + 6·17-s − 2·19-s + 5·23-s − 27-s − 5·29-s − 4·31-s − 5·33-s − 37-s + 2·39-s + 12·41-s + 5·43-s − 2·47-s − 6·51-s − 14·53-s + 2·57-s + 2·59-s − 5·67-s − 5·69-s + 9·71-s + 10·73-s − 11·79-s + 81-s − 16·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.50·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.04·23-s − 0.192·27-s − 0.928·29-s − 0.718·31-s − 0.870·33-s − 0.164·37-s + 0.320·39-s + 1.87·41-s + 0.762·43-s − 0.291·47-s − 0.840·51-s − 1.92·53-s + 0.264·57-s + 0.260·59-s − 0.610·67-s − 0.601·69-s + 1.06·71-s + 1.17·73-s − 1.23·79-s + 1/9·81-s − 1.75·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.227659882\)
\(L(\frac12)\) \(\approx\) \(2.227659882\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.36004665465433, −14.07160509640852, −13.14250173273696, −12.62749222659804, −12.41970299874057, −11.80543580404813, −11.15676506996117, −11.03185614251765, −10.24452050418790, −9.573641297564021, −9.356818545846574, −8.803382508489015, −7.960274844082900, −7.466676840246762, −7.016380996335149, −6.366995006295682, −5.874665126911981, −5.350930004223713, −4.686799794982572, −4.085260898434934, −3.531465115247838, −2.861493860917926, −1.900914084625383, −1.275776977248174, −0.5787647838126680, 0.5787647838126680, 1.275776977248174, 1.900914084625383, 2.861493860917926, 3.531465115247838, 4.085260898434934, 4.686799794982572, 5.350930004223713, 5.874665126911981, 6.366995006295682, 7.016380996335149, 7.466676840246762, 7.960274844082900, 8.803382508489015, 9.356818545846574, 9.573641297564021, 10.24452050418790, 11.03185614251765, 11.15676506996117, 11.80543580404813, 12.41970299874057, 12.62749222659804, 13.14250173273696, 14.07160509640852, 14.36004665465433

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