Properties

Label 2-15600-1.1-c1-0-43
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 − 4·7-s + 9-s − 13-s − 5·19-s − 4·21-s + 27-s + 3·29-s + 4·31-s + 7·37-s − 39-s + 3·41-s + 2·43-s + 9·47-s + 9·49-s − 9·53-s − 5·57-s − 6·59-s + 8·61-s − 4·63-s + 5·67-s + 3·71-s + 4·73-s − 11·79-s + 81-s − 6·83-s + 3·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 1.14·19-s − 0.872·21-s + 0.192·27-s + 0.557·29-s + 0.718·31-s + 1.15·37-s − 0.160·39-s + 0.468·41-s + 0.304·43-s + 1.31·47-s + 9/7·49-s − 1.23·53-s − 0.662·57-s − 0.781·59-s + 1.02·61-s − 0.503·63-s + 0.610·67-s + 0.356·71-s + 0.468·73-s − 1.23·79-s + 1/9·81-s − 0.658·83-s + 0.321·87-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.09793182699033, −15.70275471840298, −15.32521153016333, −14.48273778530153, −14.17812630052492, −13.35391188277604, −12.99422462112745, −12.51441110483801, −12.00597696594013, −11.10357669564575, −10.53608143059973, −9.874821125277884, −9.542338247348030, −8.885473896736737, −8.315312491543125, −7.610067293275028, −6.950718013189171, −6.323299685874655, −5.956694746323994, −4.881094293332312, −4.173561952074681, −3.588136451985782, −2.718314028682113, −2.385137556069775, −1.076740622380979, 0, 1.076740622380979, 2.385137556069775, 2.718314028682113, 3.588136451985782, 4.173561952074681, 4.881094293332312, 5.956694746323994, 6.323299685874655, 6.950718013189171, 7.610067293275028, 8.315312491543125, 8.885473896736737, 9.542338247348030, 9.874821125277884, 10.53608143059973, 11.10357669564575, 12.00597696594013, 12.51441110483801, 12.99422462112745, 13.35391188277604, 14.17812630052492, 14.48273778530153, 15.32521153016333, 15.70275471840298, 16.09793182699033

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