Properties

Label 2-53312-1.1-c1-0-42
Degree $2$
Conductor $53312$
Sign $-1$
Analytic cond. $425.698$
Root an. cond. $20.6324$
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  − 2·3-s + 3·5-s + 9-s + 2·13-s − 6·15-s + 17-s − 7·19-s − 3·23-s + 4·25-s + 4·27-s + 6·29-s + 4·31-s + 7·37-s − 4·39-s − 6·41-s + 43-s + 3·45-s − 2·51-s + 6·53-s + 14·57-s + 9·59-s + 14·61-s + 6·65-s − 5·67-s + 6·69-s − 15·71-s − 2·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.34·5-s + 1/3·9-s + 0.554·13-s − 1.54·15-s + 0.242·17-s − 1.60·19-s − 0.625·23-s + 4/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.15·37-s − 0.640·39-s − 0.937·41-s + 0.152·43-s + 0.447·45-s − 0.280·51-s + 0.824·53-s + 1.85·57-s + 1.17·59-s + 1.79·61-s + 0.744·65-s − 0.610·67-s + 0.722·69-s − 1.78·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(425.698\)
Root analytic conductor: \(20.6324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−14.59104617191240, −14.18941988972573, −13.59561978843735, −13.12187282604513, −12.73040704651800, −12.11964343988004, −11.52943069615175, −11.22323468119281, −10.42980932955041, −10.08601363241038, −9.950281965600197, −8.824235264365414, −8.640522307119708, −8.001427426131667, −6.996531034005776, −6.551638056409799, −6.167176281216724, −5.642613902916327, −5.328583734921397, −4.398670940885561, −4.133395153423772, −2.911576731573686, −2.442923746372828, −1.610537967150409, −0.9711219220449954, 0, 0.9711219220449954, 1.610537967150409, 2.442923746372828, 2.911576731573686, 4.133395153423772, 4.398670940885561, 5.328583734921397, 5.642613902916327, 6.167176281216724, 6.551638056409799, 6.996531034005776, 8.001427426131667, 8.640522307119708, 8.824235264365414, 9.950281965600197, 10.08601363241038, 10.42980932955041, 11.22323468119281, 11.52943069615175, 12.11964343988004, 12.73040704651800, 13.12187282604513, 13.59561978843735, 14.18941988972573, 14.59104617191240

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