Properties

Label 2-15600-1.1-c1-0-44
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 − 4·11-s − 13-s − 6·17-s − 4·21-s − 4·23-s − 27-s − 6·29-s + 8·31-s + 4·33-s + 2·37-s + 39-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s + 6·51-s + 2·53-s − 4·59-s + 14·61-s + 4·63-s − 12·67-s + 4·69-s + 8·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-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 + 0.840·51-s + 0.274·53-s − 0.520·59-s + 1.79·61-s + 0.503·63-s − 1.46·67-s + 0.481·69-s + 0.949·71-s + 1.17·73-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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−16.07831762894163, −15.85532532410906, −15.06683541459805, −14.84872017633285, −13.91805839754785, −13.58476597497077, −12.90900076265935, −12.33296780280895, −11.61794539439106, −11.25976822527705, −10.72165843714871, −10.26958327197170, −9.482265624478791, −8.764066898141489, −8.072282832666461, −7.732125439267761, −7.057550250724624, −6.235235609616548, −5.595588634825185, −4.971033079245870, −4.505689113583437, −3.867763825407368, −2.457957069788756, −2.195296423127552, −1.081175386244187, 0, 1.081175386244187, 2.195296423127552, 2.457957069788756, 3.867763825407368, 4.505689113583437, 4.971033079245870, 5.595588634825185, 6.235235609616548, 7.057550250724624, 7.732125439267761, 8.072282832666461, 8.764066898141489, 9.482265624478791, 10.26958327197170, 10.72165843714871, 11.25976822527705, 11.61794539439106, 12.33296780280895, 12.90900076265935, 13.58476597497077, 13.91805839754785, 14.84872017633285, 15.06683541459805, 15.85532532410906, 16.07831762894163

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