Properties

Label 2-15600-1.1-c1-0-29
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 + 3·7-s + 9-s + 3·11-s − 13-s − 3·17-s + 3·21-s + 4·23-s + 27-s + 5·29-s + 3·31-s + 3·33-s + 12·37-s − 39-s + 2·41-s + 4·43-s + 3·47-s + 2·49-s − 3·51-s − 9·53-s − 15·59-s − 3·61-s + 3·63-s − 7·67-s + 4·69-s + 8·71-s + 16·73-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 + 0.834·23-s + 0.192·27-s + 0.928·29-s + 0.538·31-s + 0.522·33-s + 1.97·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.437·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s − 1.95·59-s − 0.384·61-s + 0.377·63-s − 0.855·67-s + 0.481·69-s + 0.949·71-s + 1.87·73-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

\(L(1)\) \(\approx\) \(3.793168723\)
\(L(\frac12)\) \(\approx\) \(3.793168723\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−15.86833409462489, −15.28822088171355, −14.87542180748219, −14.31713101878645, −13.93256655382890, −13.36809587776098, −12.57654910018195, −12.17288691919088, −11.34735362482457, −11.06987605099880, −10.42018833110076, −9.381329292038862, −9.330446064789017, −8.492203624377703, −7.940667026619036, −7.496488158047808, −6.606907150872393, −6.208344411770715, −5.151888959964007, −4.518214649358876, −4.181811270700485, −3.108914828304875, −2.466611741804090, −1.607649197267606, −0.8775091327183000, 0.8775091327183000, 1.607649197267606, 2.466611741804090, 3.108914828304875, 4.181811270700485, 4.518214649358876, 5.151888959964007, 6.208344411770715, 6.606907150872393, 7.496488158047808, 7.940667026619036, 8.492203624377703, 9.330446064789017, 9.381329292038862, 10.42018833110076, 11.06987605099880, 11.34735362482457, 12.17288691919088, 12.57654910018195, 13.36809587776098, 13.93256655382890, 14.31713101878645, 14.87542180748219, 15.28822088171355, 15.86833409462489

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