Properties

Label 2-31200-1.1-c1-0-41
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
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 − 2·17-s − 8·19-s − 4·21-s + 4·23-s − 27-s + 6·29-s + 8·31-s − 6·37-s + 39-s − 6·41-s − 4·43-s + 4·47-s + 9·49-s + 2·51-s − 6·53-s + 8·57-s − 8·59-s − 10·61-s + 4·63-s + 4·67-s − 4·69-s − 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s − 1.83·19-s − 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s + 1.05·57-s − 1.04·59-s − 1.28·61-s + 0.503·63-s + 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.39609463636234, −14.89687053372113, −14.35032706006254, −13.71961065017036, −13.36903249324583, −12.44816045958425, −12.21065768785155, −11.63313347950783, −10.96720840594558, −10.70681121555071, −10.23405958268607, −9.412019409272952, −8.655667883605811, −8.344565810189309, −7.815247415460409, −6.972654748885151, −6.557925304922014, −5.962322403252525, −5.055489123995349, −4.695307301625142, −4.382307750193421, −3.356954747131594, −2.430899552878592, −1.808301980662715, −1.072801961514878, 0, 1.072801961514878, 1.808301980662715, 2.430899552878592, 3.356954747131594, 4.382307750193421, 4.695307301625142, 5.055489123995349, 5.962322403252525, 6.557925304922014, 6.972654748885151, 7.815247415460409, 8.344565810189309, 8.655667883605811, 9.412019409272952, 10.23405958268607, 10.70681121555071, 10.96720840594558, 11.63313347950783, 12.21065768785155, 12.44816045958425, 13.36903249324583, 13.71961065017036, 14.35032706006254, 14.89687053372113, 15.39609463636234

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