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Properties

Label 2-203280-1.1-c1-0-76
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 − 4·19-s − 21-s + 25-s − 27-s + 6·29-s + 4·31-s − 35-s + 2·37-s + 2·39-s − 6·41-s + 8·43-s − 45-s + 12·47-s + 49-s − 6·51-s + 6·53-s + 4·57-s + 12·59-s − 2·61-s + 63-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 − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.125·63-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

\(L(1)\) \(\approx\) \(2.400542703\)
\(L(\frac12)\) \(\approx\) \(2.400542703\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.77103409012892, −12.60660444064054, −11.93872309159885, −11.77700389438119, −11.34523354356885, −10.51651995770405, −10.27600014397820, −10.08386847197406, −9.217894290044351, −8.694786117537490, −8.365692139180543, −7.596203153641091, −7.427142543492606, −6.889765144091092, −6.084494387261703, −5.915107521363603, −5.201802954637214, −4.706586458886515, −4.284462252077547, −3.733650384950501, −3.011478007075352, −2.479702541979139, −1.777384211804785, −0.9251716465592876, −0.5759057294369549, 0.5759057294369549, 0.9251716465592876, 1.777384211804785, 2.479702541979139, 3.011478007075352, 3.733650384950501, 4.284462252077547, 4.706586458886515, 5.201802954637214, 5.915107521363603, 6.084494387261703, 6.889765144091092, 7.427142543492606, 7.596203153641091, 8.365692139180543, 8.694786117537490, 9.217894290044351, 10.08386847197406, 10.27600014397820, 10.51651995770405, 11.34523354356885, 11.77700389438119, 11.93872309159885, 12.60660444064054, 12.77103409012892

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