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Properties

Label 2-203280-1.1-c1-0-64
Degree $2$
Conductor $203280$
Sign $1$
Analytic cond. $1623.19$
Root an. cond. $40.2889$
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 − 5-s + 7-s + 9-s + 4·13-s + 15-s − 3·17-s − 19-s − 21-s + 3·23-s + 25-s − 27-s + 9·29-s + 10·31-s − 35-s − 4·37-s − 4·39-s − 6·41-s − 43-s − 45-s + 6·47-s + 49-s + 3·51-s − 9·53-s + 57-s − 15·59-s + 13·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.258·15-s − 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 1.79·31-s − 0.169·35-s − 0.657·37-s − 0.640·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.420·51-s − 1.23·53-s + 0.132·57-s − 1.95·59-s + 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203280\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1623.19\)
Root analytic conductor: \(40.2889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.405483128\)
\(L(\frac12)\) \(\approx\) \(2.405483128\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−12.96286194256673, −12.52065704809711, −12.00732355872815, −11.68201695710189, −11.17179115065794, −10.66606781261077, −10.53021220567874, −9.796266353192032, −9.274078592182557, −8.621758271662120, −8.292322777079140, −7.997114395846543, −7.168992209858968, −6.631429775838324, −6.462434770063384, −5.838278240195125, −5.144752740394695, −4.629114351563909, −4.419740363612155, −3.600279519485186, −3.155253066071919, −2.433788785441718, −1.717114689402437, −1.008555987614650, −0.5453500618579272, 0.5453500618579272, 1.008555987614650, 1.717114689402437, 2.433788785441718, 3.155253066071919, 3.600279519485186, 4.419740363612155, 4.629114351563909, 5.144752740394695, 5.838278240195125, 6.462434770063384, 6.631429775838324, 7.168992209858968, 7.997114395846543, 8.292322777079140, 8.621758271662120, 9.274078592182557, 9.796266353192032, 10.53021220567874, 10.66606781261077, 11.17179115065794, 11.68201695710189, 12.00732355872815, 12.52065704809711, 12.96286194256673

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