Properties

Label 2-31200-1.1-c1-0-43
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 − 6·19-s − 2·21-s + 6·23-s + 27-s − 8·29-s + 4·33-s − 10·37-s − 39-s − 10·41-s + 12·43-s + 12·47-s − 3·49-s + 6·53-s − 6·57-s + 12·59-s − 10·61-s − 2·63-s + 4·67-s + 6·69-s − 8·71-s + 4·73-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 − 1.37·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.48·29-s + 0.696·33-s − 1.64·37-s − 0.160·39-s − 1.56·41-s + 1.82·43-s + 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s + 0.468·73-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 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.28095865238890, −14.85353376096094, −14.31016012143633, −13.85438597415370, −13.08869830472415, −12.90047334149188, −12.21439176918458, −11.75879842578572, −10.99066756665578, −10.50318838051784, −9.950815304139681, −9.235885036069966, −8.868263318338495, −8.614860347317702, −7.515648162112321, −7.166108472756470, −6.598442126060236, −6.044033499826265, −5.292640948821192, −4.533237975986246, −3.759532972298341, −3.545800991506883, −2.573010789020403, −1.972639566682210, −1.094085955791147, 0, 1.094085955791147, 1.972639566682210, 2.573010789020403, 3.545800991506883, 3.759532972298341, 4.533237975986246, 5.292640948821192, 6.044033499826265, 6.598442126060236, 7.166108472756470, 7.515648162112321, 8.614860347317702, 8.868263318338495, 9.235885036069966, 9.950815304139681, 10.50318838051784, 10.99066756665578, 11.75879842578572, 12.21439176918458, 12.90047334149188, 13.08869830472415, 13.85438597415370, 14.31016012143633, 14.85353376096094, 15.28095865238890

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