Properties

Label 2-31200-1.1-c1-0-16
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 + 4·7-s + 9-s + 13-s + 3·19-s − 4·21-s − 4·23-s − 27-s − 29-s + 8·31-s + 37-s − 39-s + 41-s + 6·43-s − 11·47-s + 9·49-s + 3·53-s − 3·57-s − 10·59-s + 4·61-s + 4·63-s + 13·67-s + 4·69-s − 9·71-s + 3·79-s + 81-s + 2·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.688·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.164·37-s − 0.160·39-s + 0.156·41-s + 0.914·43-s − 1.60·47-s + 9/7·49-s + 0.412·53-s − 0.397·57-s − 1.30·59-s + 0.512·61-s + 0.503·63-s + 1.58·67-s + 0.481·69-s − 1.06·71-s + 0.337·79-s + 1/9·81-s + 0.219·83-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

\(L(1)\) \(\approx\) \(2.504288771\)
\(L(\frac12)\) \(\approx\) \(2.504288771\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.09339555993224, −14.37753459373410, −14.25309251060685, −13.46758166067936, −13.07134560092111, −12.15034913539542, −11.88053756287980, −11.43948284396602, −10.84117525059029, −10.45790093915670, −9.714674201899500, −9.236934125619086, −8.309497205182981, −8.076220395917364, −7.516421195085817, −6.778391028594407, −6.151348829356475, −5.550526821425567, −4.966092408505166, −4.478212088058883, −3.860338910721244, −2.941867870177652, −2.064379901147581, −1.415341196941911, −0.6687949435050789, 0.6687949435050789, 1.415341196941911, 2.064379901147581, 2.941867870177652, 3.860338910721244, 4.478212088058883, 4.966092408505166, 5.550526821425567, 6.151348829356475, 6.778391028594407, 7.516421195085817, 8.076220395917364, 8.309497205182981, 9.236934125619086, 9.714674201899500, 10.45790093915670, 10.84117525059029, 11.43948284396602, 11.88053756287980, 12.15034913539542, 13.07134560092111, 13.46758166067936, 14.25309251060685, 14.37753459373410, 15.09339555993224

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