Properties

Label 2-58800-1.1-c1-0-111
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 − 4·17-s + 6·19-s + 23-s − 27-s + 29-s + 2·31-s + 5·33-s + 5·37-s + 2·39-s + 7·43-s − 2·47-s + 4·51-s − 6·53-s − 6·57-s − 6·59-s − 3·67-s − 69-s − 9·71-s + 12·73-s − 3·79-s + 81-s + 6·83-s − 87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s + 0.208·23-s − 0.192·27-s + 0.185·29-s + 0.359·31-s + 0.870·33-s + 0.821·37-s + 0.320·39-s + 1.06·43-s − 0.291·47-s + 0.560·51-s − 0.824·53-s − 0.794·57-s − 0.781·59-s − 0.366·67-s − 0.120·69-s − 1.06·71-s + 1.40·73-s − 0.337·79-s + 1/9·81-s + 0.658·83-s − 0.107·87-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.58735403829794, −14.02097603908343, −13.44305310923759, −13.10766587642502, −12.55693167003464, −12.07338705921824, −11.48699150908229, −11.03936897579808, −10.53812175572242, −10.09311985690521, −9.447585133918911, −9.099227362644718, −8.208310013141925, −7.729939966870240, −7.384585335744920, −6.682473705844853, −6.108185897615886, −5.492916700597101, −4.978684568975749, −4.608150108180575, −3.829776851115951, −2.873464837110734, −2.617250102460411, −1.691804586675551, −0.7792044067789580, 0, 0.7792044067789580, 1.691804586675551, 2.617250102460411, 2.873464837110734, 3.829776851115951, 4.608150108180575, 4.978684568975749, 5.492916700597101, 6.108185897615886, 6.682473705844853, 7.384585335744920, 7.729939966870240, 8.208310013141925, 9.099227362644718, 9.447585133918911, 10.09311985690521, 10.53812175572242, 11.03936897579808, 11.48699150908229, 12.07338705921824, 12.55693167003464, 13.10766587642502, 13.44305310923759, 14.02097603908343, 14.58735403829794

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