Properties

Label 2-20328-1.1-c1-0-25
Degree $2$
Conductor $20328$
Sign $-1$
Analytic cond. $162.319$
Root an. cond. $12.7404$
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 + 8·17-s − 5·19-s + 21-s − 4·25-s + 27-s − 5·29-s − 6·31-s + 35-s − 11·37-s − 39-s − 8·43-s + 45-s + 7·47-s + 49-s + 8·51-s + 2·53-s − 5·57-s + 7·59-s + 2·61-s + 63-s − 65-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 + 1.94·17-s − 1.14·19-s + 0.218·21-s − 4/5·25-s + 0.192·27-s − 0.928·29-s − 1.07·31-s + 0.169·35-s − 1.80·37-s − 0.160·39-s − 1.21·43-s + 0.149·45-s + 1.02·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.662·57-s + 0.911·59-s + 0.256·61-s + 0.125·63-s − 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20328\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(162.319\)
Root analytic conductor: \(12.7404\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{20328} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 20328,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.88699690734641, −15.19034236339653, −14.60325562272202, −14.53677392222326, −13.70907336378558, −13.32312317877219, −12.66440361896692, −12.08255481542023, −11.70039398061087, −10.65953130888043, −10.45192079503806, −9.737955353646916, −9.268103769147518, −8.579053192913708, −8.079633988355017, −7.438210360662755, −6.982663364064073, −6.077052248082950, −5.458948152983378, −5.041168515524569, −3.921939535369083, −3.636423356802022, −2.687382895711676, −1.914620511185696, −1.380206938105230, 0, 1.380206938105230, 1.914620511185696, 2.687382895711676, 3.636423356802022, 3.921939535369083, 5.041168515524569, 5.458948152983378, 6.077052248082950, 6.982663364064073, 7.438210360662755, 8.079633988355017, 8.579053192913708, 9.268103769147518, 9.737955353646916, 10.45192079503806, 10.65953130888043, 11.70039398061087, 12.08255481542023, 12.66440361896692, 13.32312317877219, 13.70907336378558, 14.53677392222326, 14.60325562272202, 15.19034236339653, 15.88699690734641

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