Properties

Label 2-31200-1.1-c1-0-53
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 − 2·7-s + 9-s + 4·11-s − 13-s + 6·17-s + 6·19-s − 2·21-s + 27-s − 2·29-s − 6·31-s + 4·33-s − 10·37-s − 39-s + 8·41-s − 12·43-s − 12·47-s − 3·49-s + 6·51-s + 6·53-s + 6·57-s + 2·61-s − 2·63-s − 2·67-s − 8·71-s − 14·73-s − 8·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 1.45·17-s + 1.37·19-s − 0.436·21-s + 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.696·33-s − 1.64·37-s − 0.160·39-s + 1.24·41-s − 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s + 0.794·57-s + 0.256·61-s − 0.251·63-s − 0.244·67-s − 0.949·71-s − 1.63·73-s − 0.911·77-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.08261547453988, −14.85862558382650, −14.28426328236115, −13.87134632314229, −13.29815224401764, −12.69898310005212, −12.19528361987402, −11.72693195023186, −11.22324526242593, −10.26209521299586, −9.890280197992809, −9.468753584100077, −8.972815221739293, −8.320959048148488, −7.665710865424349, −7.077197518059167, −6.739439115904197, −5.829689018019913, −5.402170563243284, −4.614422152866551, −3.664733597417047, −3.430750452974926, −2.854093961311423, −1.689929225430157, −1.251835092022452, 0, 1.251835092022452, 1.689929225430157, 2.854093961311423, 3.430750452974926, 3.664733597417047, 4.614422152866551, 5.402170563243284, 5.829689018019913, 6.739439115904197, 7.077197518059167, 7.665710865424349, 8.320959048148488, 8.972815221739293, 9.468753584100077, 9.890280197992809, 10.26209521299586, 11.22324526242593, 11.72693195023186, 12.19528361987402, 12.69898310005212, 13.29815224401764, 13.87134632314229, 14.28426328236115, 14.85862558382650, 15.08261547453988

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