Properties

Label 2-31200-1.1-c1-0-9
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

\(L(1)\) \(\approx\) \(2.015913573\)
\(L(\frac12)\) \(\approx\) \(2.015913573\)
\(L(\frac{3}{2})\) 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

−15.19602020734685, −14.56594799996830, −13.93239938043277, −13.38492170882549, −13.22170585661998, −12.32734791325939, −12.19408266394184, −11.40274457201970, −10.65032741617791, −10.09373475535197, −9.755772068495470, −9.047345337366105, −8.844008621596696, −7.920175345517748, −7.401039874375451, −6.855616901608014, −6.294302582291861, −5.722651806131351, −4.896636766249136, −4.271755341690153, −3.455011427347306, −3.013305232245842, −2.513789108180572, −1.445038150729997, −0.5316985626309652, 0.5316985626309652, 1.445038150729997, 2.513789108180572, 3.013305232245842, 3.455011427347306, 4.271755341690153, 4.896636766249136, 5.722651806131351, 6.294302582291861, 6.855616901608014, 7.401039874375451, 7.920175345517748, 8.844008621596696, 9.047345337366105, 9.755772068495470, 10.09373475535197, 10.65032741617791, 11.40274457201970, 12.19408266394184, 12.32734791325939, 13.22170585661998, 13.38492170882549, 13.93239938043277, 14.56594799996830, 15.19602020734685

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