Properties

Label 2-31200-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·7-s + 9-s + 3·11-s − 13-s − 7·17-s + 8·19-s − 3·21-s + 4·23-s − 27-s − 3·29-s + 11·31-s − 3·33-s + 39-s − 2·41-s − 8·43-s − 9·47-s + 2·49-s + 7·51-s − 9·53-s − 8·57-s + 9·59-s + 61-s + 3·63-s + 5·67-s − 4·69-s + 12·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s + 1.83·19-s − 0.654·21-s + 0.834·23-s − 0.192·27-s − 0.557·29-s + 1.97·31-s − 0.522·33-s + 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.31·47-s + 2/7·49-s + 0.980·51-s − 1.23·53-s − 1.05·57-s + 1.17·59-s + 0.128·61-s + 0.377·63-s + 0.610·67-s − 0.481·69-s + 1.40·73-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: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.419586782\)
\(L(\frac12)\) \(\approx\) \(2.419586782\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.20790738356953, −14.52551548375905, −14.06067384987697, −13.50817773843415, −13.05914763346936, −12.27287018601085, −11.67246248165022, −11.41064196833652, −11.12387944524712, −10.28817511650346, −9.648973761534161, −9.276072103488369, −8.451369138717322, −8.092826654524602, −7.318027829036732, −6.707166968950704, −6.411999594517211, −5.441276562925863, −4.836153732505320, −4.672188257253620, −3.746676549502182, −3.012331015833255, −2.053478930821129, −1.408512980975595, −0.6584186589835133, 0.6584186589835133, 1.408512980975595, 2.053478930821129, 3.012331015833255, 3.746676549502182, 4.672188257253620, 4.836153732505320, 5.441276562925863, 6.411999594517211, 6.707166968950704, 7.318027829036732, 8.092826654524602, 8.451369138717322, 9.276072103488369, 9.648973761534161, 10.28817511650346, 11.12387944524712, 11.41064196833652, 11.67246248165022, 12.27287018601085, 13.05914763346936, 13.50817773843415, 14.06067384987697, 14.52551548375905, 15.20790738356953

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