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Properties

Label 2-58800-1.1-c1-0-203
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 + 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: \(1\)
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 \( 1 - 5 T + p T^{2} \)
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

−14.53391045994397, −14.05830002708176, −13.75930188345229, −12.92560626851072, −12.46507228137729, −11.88086614449427, −11.67988489078625, −11.29294979329258, −10.26884338073999, −10.00783152092431, −9.679284466930012, −8.932619826344146, −8.380822415058099, −7.764821393673912, −7.189089144355869, −6.696313889114809, −6.162135506428454, −5.513699765923609, −5.138358501740705, −4.340018313897982, −3.736011301572068, −3.336938870578793, −2.331738629996956, −1.544644958261419, −1.013352258278550, 0, 1.013352258278550, 1.544644958261419, 2.331738629996956, 3.336938870578793, 3.736011301572068, 4.340018313897982, 5.138358501740705, 5.513699765923609, 6.162135506428454, 6.696313889114809, 7.189089144355869, 7.764821393673912, 8.380822415058099, 8.932619826344146, 9.679284466930012, 10.00783152092431, 10.26884338073999, 11.29294979329258, 11.67988489078625, 11.88086614449427, 12.46507228137729, 12.92560626851072, 13.75930188345229, 14.05830002708176, 14.53391045994397

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