Properties

Label 2-31200-1.1-c1-0-42
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 − 7-s + 9-s − 11-s + 13-s + 3·17-s − 6·19-s − 21-s + 5·23-s + 27-s − 4·31-s − 33-s − 5·37-s + 39-s + 7·41-s − 6·43-s − 8·47-s − 6·49-s + 3·51-s + 3·53-s − 6·57-s + 12·59-s − 15·61-s − 63-s + 8·67-s + 5·69-s + 15·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.218·21-s + 1.04·23-s + 0.192·27-s − 0.718·31-s − 0.174·33-s − 0.821·37-s + 0.160·39-s + 1.09·41-s − 0.914·43-s − 1.16·47-s − 6/7·49-s + 0.420·51-s + 0.412·53-s − 0.794·57-s + 1.56·59-s − 1.92·61-s − 0.125·63-s + 0.977·67-s + 0.601·69-s + 1.78·71-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 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−15.20526668358096, −14.84738186845038, −14.38087296226151, −13.77388206852702, −13.14206529054804, −12.79491840906454, −12.42850948488167, −11.53994184099753, −11.09377887993833, −10.38770174299421, −10.08649044541878, −9.285577039313029, −8.913667088655283, −8.302474703620184, −7.778336635296308, −7.162209559502086, −6.510663433327409, −6.069715109931322, −5.121196747197159, −4.773206631155830, −3.692723953795958, −3.497235578723546, −2.605323269861927, −1.973400287052497, −1.093120780167786, 0, 1.093120780167786, 1.973400287052497, 2.605323269861927, 3.497235578723546, 3.692723953795958, 4.773206631155830, 5.121196747197159, 6.069715109931322, 6.510663433327409, 7.162209559502086, 7.778336635296308, 8.302474703620184, 8.913667088655283, 9.285577039313029, 10.08649044541878, 10.38770174299421, 11.09377887993833, 11.53994184099753, 12.42850948488167, 12.79491840906454, 13.14206529054804, 13.77388206852702, 14.38087296226151, 14.84738186845038, 15.20526668358096

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