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Properties

Label 2-203280-1.1-c1-0-158
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 + 2·19-s − 21-s + 3·23-s + 25-s − 27-s + 9·29-s + 7·31-s + 35-s − 4·37-s − 39-s − 3·41-s + 5·43-s + 45-s + 49-s + 6·51-s − 6·53-s − 2·57-s + 9·59-s + 7·61-s + 63-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 + 0.458·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 1.25·31-s + 0.169·35-s − 0.657·37-s − 0.160·39-s − 0.468·41-s + 0.762·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.264·57-s + 1.17·59-s + 0.896·61-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.29930413480971, −12.85625468421797, −12.23070159592630, −11.91765669001349, −11.27821289294500, −11.09133055181634, −10.41638656958419, −10.14042225444370, −9.582644195057506, −8.994572175671876, −8.487435378497260, −8.263956465170599, −7.399319859622621, −6.936618083946554, −6.539752245661739, −6.106920404680873, −5.463442554063565, −5.012306389712310, −4.477974668889747, −4.158716421121869, −3.235963200355986, −2.698206776274037, −2.145929836861495, −1.350757255816798, −0.9137061960391457, 0, 0.9137061960391457, 1.350757255816798, 2.145929836861495, 2.698206776274037, 3.235963200355986, 4.158716421121869, 4.477974668889747, 5.012306389712310, 5.463442554063565, 6.106920404680873, 6.539752245661739, 6.936618083946554, 7.399319859622621, 8.263956465170599, 8.487435378497260, 8.994572175671876, 9.582644195057506, 10.14042225444370, 10.41638656958419, 11.09133055181634, 11.27821289294500, 11.91765669001349, 12.23070159592630, 12.85625468421797, 13.29930413480971

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