Properties

Label 2-58800-1.1-c1-0-77
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 − 4·13-s − 3·17-s − 4·19-s + 3·23-s − 27-s − 6·29-s + 5·31-s + 6·33-s − 8·37-s + 4·39-s + 3·41-s + 8·43-s − 9·47-s + 3·51-s − 12·53-s + 4·57-s + 6·59-s − 2·61-s + 8·67-s − 3·69-s + 9·71-s + 14·73-s + 7·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.10·13-s − 0.727·17-s − 0.917·19-s + 0.625·23-s − 0.192·27-s − 1.11·29-s + 0.898·31-s + 1.04·33-s − 1.31·37-s + 0.640·39-s + 0.468·41-s + 1.21·43-s − 1.31·47-s + 0.420·51-s − 1.64·53-s + 0.529·57-s + 0.781·59-s − 0.256·61-s + 0.977·67-s − 0.361·69-s + 1.06·71-s + 1.63·73-s + 0.787·79-s + 1/9·81-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−14.55078087164620, −14.15951476142324, −13.35109940602199, −13.00886384799779, −12.61322311408347, −12.20450718459804, −11.39510114505876, −10.96857855110632, −10.62149787427790, −10.07793241786489, −9.520417475093207, −9.017922605100465, −8.138332378201710, −7.915698619643814, −7.243839073220488, −6.699743685002921, −6.190350914084797, −5.370574889249014, −5.027144332193383, −4.635486434160748, −3.807072159767105, −3.019476567196474, −2.312997891436848, −1.944625360141018, −0.6420638674460017, 0, 0.6420638674460017, 1.944625360141018, 2.312997891436848, 3.019476567196474, 3.807072159767105, 4.635486434160748, 5.027144332193383, 5.370574889249014, 6.190350914084797, 6.699743685002921, 7.243839073220488, 7.915698619643814, 8.138332378201710, 9.017922605100465, 9.520417475093207, 10.07793241786489, 10.62149787427790, 10.96857855110632, 11.39510114505876, 12.20450718459804, 12.61322311408347, 13.00886384799779, 13.35109940602199, 14.15951476142324, 14.55078087164620

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