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Properties

Label 2-203280-1.1-c1-0-168
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 + 2·13-s − 15-s + 2·17-s − 21-s + 25-s + 27-s + 2·29-s + 4·31-s + 35-s + 6·37-s + 2·39-s − 6·41-s + 12·43-s − 45-s − 4·47-s + 49-s + 2·51-s + 6·53-s + 4·59-s + 10·61-s − 63-s − 2·65-s + 4·67-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 − 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.125·63-s − 0.248·65-s + 0.488·67-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.25657629891981, −12.88013722196504, −12.37481242603500, −11.93234589076127, −11.38826327783140, −11.02058157390184, −10.33382621123622, −9.997415169269179, −9.534076209686840, −8.969026223913676, −8.477401952564838, −8.136783363040636, −7.648701662914280, −6.991110443947980, −6.751896116821714, −5.961119721546449, −5.644011624306439, −4.877779711613484, −4.320630222725695, −3.830172494020724, −3.409572941568142, −2.652995223684972, −2.423630828468750, −1.336372843414541, −0.9594219162270808, 0, 0.9594219162270808, 1.336372843414541, 2.423630828468750, 2.652995223684972, 3.409572941568142, 3.830172494020724, 4.320630222725695, 4.877779711613484, 5.644011624306439, 5.961119721546449, 6.751896116821714, 6.991110443947980, 7.648701662914280, 8.136783363040636, 8.477401952564838, 8.969026223913676, 9.534076209686840, 9.997415169269179, 10.33382621123622, 11.02058157390184, 11.38826327783140, 11.93234589076127, 12.37481242603500, 12.88013722196504, 13.25657629891981

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