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Properties

Label 2-203280-1.1-c1-0-60
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 − 2·17-s + 4·19-s + 21-s + 8·23-s + 25-s − 27-s + 2·29-s − 35-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s + 45-s + 49-s + 2·51-s − 10·53-s − 4·57-s − 12·59-s − 14·61-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 − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.169·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s − 1.79·61-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

\(L(1)\) \(\approx\) \(2.474643530\)
\(L(\frac12)\) \(\approx\) \(2.474643530\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.03385001251519, −12.53739393326919, −12.26781682453352, −11.48226407183122, −11.11781041915277, −10.87343229364320, −10.27989718810022, −9.652524822614309, −9.375325800322650, −8.936201179972295, −8.348486385149243, −7.611510244808812, −7.372780095966079, −6.532125272907552, −6.394630981932995, −5.854464551367737, −5.233233472172239, −4.778134154534427, −4.360104914381379, −3.481603123729205, −3.100099741163353, −2.511191452316743, −1.674691068450958, −1.109047115536723, −0.5152216768537001, 0.5152216768537001, 1.109047115536723, 1.674691068450958, 2.511191452316743, 3.100099741163353, 3.481603123729205, 4.360104914381379, 4.778134154534427, 5.233233472172239, 5.854464551367737, 6.394630981932995, 6.532125272907552, 7.372780095966079, 7.611510244808812, 8.348486385149243, 8.936201179972295, 9.375325800322650, 9.652524822614309, 10.27989718810022, 10.87343229364320, 11.11781041915277, 11.48226407183122, 12.26781682453352, 12.53739393326919, 13.03385001251519

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