Properties

Label 2-15600-1.1-c1-0-30
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 − 7-s + 9-s + 3·11-s − 13-s + 7·17-s + 4·19-s − 21-s + 9·23-s + 27-s + 8·29-s − 4·31-s + 3·33-s + 3·37-s − 39-s − 5·41-s + 2·47-s − 6·49-s + 7·51-s + 3·53-s + 4·57-s + 12·59-s − 15·61-s − 63-s − 12·67-s + 9·69-s + 15·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 1.69·17-s + 0.917·19-s − 0.218·21-s + 1.87·23-s + 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.522·33-s + 0.493·37-s − 0.160·39-s − 0.780·41-s + 0.291·47-s − 6/7·49-s + 0.980·51-s + 0.412·53-s + 0.529·57-s + 1.56·59-s − 1.92·61-s − 0.125·63-s − 1.46·67-s + 1.08·69-s + 1.78·71-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

\(L(1)\) \(\approx\) \(3.518251113\)
\(L(\frac12)\) \(\approx\) \(3.518251113\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−16.15057970022349, −15.24571129231337, −14.90874874255460, −14.28390805791021, −13.94778419649031, −13.23706443661794, −12.67126741102077, −12.08699521202885, −11.67441124737187, −10.88596494059348, −10.15570389900707, −9.731724321358899, −9.141378823429718, −8.672683810863195, −7.856731615083818, −7.339009905851648, −6.773503707844457, −6.088269638466360, −5.238895873346774, −4.742861322066966, −3.718986663233474, −3.239120239860574, −2.676652073714998, −1.447012692165350, −0.8840017279395159, 0.8840017279395159, 1.447012692165350, 2.676652073714998, 3.239120239860574, 3.718986663233474, 4.742861322066966, 5.238895873346774, 6.088269638466360, 6.773503707844457, 7.339009905851648, 7.856731615083818, 8.672683810863195, 9.141378823429718, 9.731724321358899, 10.15570389900707, 10.88596494059348, 11.67441124737187, 12.08699521202885, 12.67126741102077, 13.23706443661794, 13.94778419649031, 14.28390805791021, 14.90874874255460, 15.24571129231337, 16.15057970022349

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