Properties

Label 2-20328-1.1-c1-0-27
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 + 2·5-s + 7-s + 9-s − 2·13-s + 2·15-s − 4·17-s + 4·19-s + 21-s − 6·23-s − 25-s + 27-s − 2·29-s − 2·31-s + 2·35-s + 2·37-s − 2·39-s + 6·43-s + 2·45-s − 12·47-s + 49-s − 4·51-s + 4·57-s − 12·59-s + 2·61-s + 63-s − 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.970·17-s + 0.917·19-s + 0.218·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.914·43-s + 0.298·45-s − 1.75·47-s + 1/7·49-s − 0.560·51-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.125·63-s − 0.496·65-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−15.84467616166909, −15.36443215165906, −14.61924841181831, −14.33990741152797, −13.65222595637043, −13.44807034942113, −12.70757036494624, −12.15693776611271, −11.47734631887970, −10.95638145673792, −10.20160575785454, −9.703898181785467, −9.337022155089819, −8.698021131922881, −7.958058948819575, −7.557583792553246, −6.817514120284830, −6.125743935714530, −5.579999256982547, −4.832884873776063, −4.239372028585566, −3.432851178120170, −2.594018298394860, −2.015244658155769, −1.389113438500676, 0, 1.389113438500676, 2.015244658155769, 2.594018298394860, 3.432851178120170, 4.239372028585566, 4.832884873776063, 5.579999256982547, 6.125743935714530, 6.817514120284830, 7.557583792553246, 7.958058948819575, 8.698021131922881, 9.337022155089819, 9.703898181785467, 10.20160575785454, 10.95638145673792, 11.47734631887970, 12.15693776611271, 12.70757036494624, 13.44807034942113, 13.65222595637043, 14.33990741152797, 14.61924841181831, 15.36443215165906, 15.84467616166909

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