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Properties

Label 2-203280-1.1-c1-0-102
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 + 2·13-s − 15-s + 6·17-s − 21-s + 8·23-s + 25-s + 27-s − 10·29-s + 8·31-s + 35-s + 2·37-s + 2·39-s + 2·41-s + 8·43-s − 45-s − 4·47-s + 49-s + 6·51-s + 10·53-s − 4·59-s + 6·61-s − 63-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.169·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.840·51-s + 1.37·53-s − 0.520·59-s + 0.768·61-s − 0.125·63-s − 0.248·65-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

\(L(1)\) \(\approx\) \(4.323192706\)
\(L(\frac12)\) \(\approx\) \(4.323192706\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.94034702658521, −12.76056810632209, −12.16038755189002, −11.64171191871905, −11.13153408122893, −10.80846345119416, −10.08077773808187, −9.765550304342156, −9.258224929973161, −8.733909419891432, −8.390247435239816, −7.710295848918768, −7.386094679788271, −7.016793168085257, −6.188938872988134, −5.906823610734125, −5.149886651459206, −4.764335638045171, −3.920161294639007, −3.633170216702340, −3.106931220719489, −2.585840479480353, −1.869401236685107, −0.9790498365632701, −0.7017470317035521, 0.7017470317035521, 0.9790498365632701, 1.869401236685107, 2.585840479480353, 3.106931220719489, 3.633170216702340, 3.920161294639007, 4.764335638045171, 5.149886651459206, 5.906823610734125, 6.188938872988134, 7.016793168085257, 7.386094679788271, 7.710295848918768, 8.390247435239816, 8.733909419891432, 9.258224929973161, 9.765550304342156, 10.08077773808187, 10.80846345119416, 11.13153408122893, 11.64171191871905, 12.16038755189002, 12.76056810632209, 12.94034702658521

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