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Properties

Label 2-203280-1.1-c1-0-103
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s + 6·17-s − 8·19-s + 21-s + 3·23-s + 25-s − 27-s + 3·29-s − 5·31-s + 35-s + 8·37-s + 39-s + 9·41-s − 5·43-s − 45-s − 6·47-s + 49-s − 6·51-s + 6·53-s + 8·57-s − 3·59-s − 61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s − 0.898·31-s + 0.169·35-s + 1.31·37-s + 0.160·39-s + 1.40·41-s − 0.762·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 1.05·57-s − 0.390·59-s − 0.128·61-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: \(1\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−12.97795302188621, −12.84566976949697, −12.36765528923400, −11.92629702954834, −11.39727490855930, −10.92348816567829, −10.58406548016934, −9.983573022861557, −9.658680641274917, −9.038205612120572, −8.517829873219521, −7.982159409596857, −7.577767566935760, −6.968969438086057, −6.587647650381973, −5.978026084618360, −5.613407229913654, −4.987869818348169, −4.374973397345173, −4.055618171081598, −3.329278622556553, −2.819017320753006, −2.151083239547498, −1.360820059443662, −0.7068630874252268, 0, 0.7068630874252268, 1.360820059443662, 2.151083239547498, 2.819017320753006, 3.329278622556553, 4.055618171081598, 4.374973397345173, 4.987869818348169, 5.613407229913654, 5.978026084618360, 6.587647650381973, 6.968969438086057, 7.577767566935760, 7.982159409596857, 8.517829873219521, 9.038205612120572, 9.658680641274917, 9.983573022861557, 10.58406548016934, 10.92348816567829, 11.39727490855930, 11.92629702954834, 12.36765528923400, 12.84566976949697, 12.97795302188621

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