Properties

Label 2-15600-1.1-c1-0-28
Degree $2$
Conductor $15600$
Sign $1$
Analytic cond. $124.566$
Root an. cond. $11.1609$
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 + 4·7-s + 9-s + 4·11-s − 13-s − 2·17-s + 8·19-s − 4·21-s − 27-s + 6·29-s + 4·31-s − 4·33-s + 2·37-s + 39-s − 10·41-s + 4·43-s + 8·47-s + 9·49-s + 2·51-s + 10·53-s − 8·57-s − 4·59-s − 2·61-s + 4·63-s − 16·67-s + 8·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.872·21-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 1.05·57-s − 0.520·59-s − 0.256·61-s + 0.503·63-s − 1.95·67-s + 0.949·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(124.566\)
Root analytic conductor: \(11.1609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.846655147\)
\(L(\frac12)\) \(\approx\) \(2.846655147\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−16.03020536668995, −15.37754228657386, −14.95133939215020, −14.25630784311151, −13.83778284641212, −13.44960979537991, −12.29397000347378, −11.94610989334890, −11.63997380665377, −11.08911358402335, −10.37000249344876, −9.879081791767015, −8.998626010746727, −8.683367078746478, −7.754696146287711, −7.383110755798420, −6.650738213349462, −5.987426046291940, −5.234307692212765, −4.734579702002488, −4.192721668837997, −3.299165328597083, −2.299643408764512, −1.383851677647705, −0.8712609500282515, 0.8712609500282515, 1.383851677647705, 2.299643408764512, 3.299165328597083, 4.192721668837997, 4.734579702002488, 5.234307692212765, 5.987426046291940, 6.650738213349462, 7.383110755798420, 7.754696146287711, 8.683367078746478, 8.998626010746727, 9.879081791767015, 10.37000249344876, 11.08911358402335, 11.63997380665377, 11.94610989334890, 12.29397000347378, 13.44960979537991, 13.83778284641212, 14.25630784311151, 14.95133939215020, 15.37754228657386, 16.03020536668995

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