Properties

Label 2-15600-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·7-s + 9-s − 3·11-s + 13-s + 3·17-s − 3·21-s − 3·23-s + 27-s + 8·29-s − 4·31-s − 3·33-s + 37-s + 39-s − 3·41-s + 4·43-s + 10·47-s + 2·49-s + 3·51-s − 9·53-s − 4·59-s + 9·61-s − 3·63-s − 4·67-s − 3·69-s − 7·71-s − 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.654·21-s − 0.625·23-s + 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.522·33-s + 0.164·37-s + 0.160·39-s − 0.468·41-s + 0.609·43-s + 1.45·47-s + 2/7·49-s + 0.420·51-s − 1.23·53-s − 0.520·59-s + 1.15·61-s − 0.377·63-s − 0.488·67-s − 0.361·69-s − 0.830·71-s − 0.702·73-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: \(1\)
Selberg data: \((2,\ 15600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−16.10591036071937, −15.81248106354314, −15.30621542470686, −14.48987367985963, −14.09840727785148, −13.42403050270740, −13.05044977335105, −12.39051022287205, −12.06175662012692, −11.10849257872200, −10.42210838372031, −10.07191466268516, −9.489241689755455, −8.850065587523136, −8.249032518955277, −7.629243166718028, −7.090518365392122, −6.281357944632358, −5.818760638433299, −4.996959574541512, −4.199854708957315, −3.439020855314875, −2.923178447925555, −2.239289723760594, −1.121785889743790, 0, 1.121785889743790, 2.239289723760594, 2.923178447925555, 3.439020855314875, 4.199854708957315, 4.996959574541512, 5.818760638433299, 6.281357944632358, 7.090518365392122, 7.629243166718028, 8.249032518955277, 8.850065587523136, 9.489241689755455, 10.07191466268516, 10.42210838372031, 11.10849257872200, 12.06175662012692, 12.39051022287205, 13.05044977335105, 13.42403050270740, 14.09840727785148, 14.48987367985963, 15.30621542470686, 15.81248106354314, 16.10591036071937

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