Properties

Label 2-54208-1.1-c1-0-66
Degree $2$
Conductor $54208$
Sign $-1$
Analytic cond. $432.853$
Root an. cond. $20.8051$
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  − 2·5-s + 7-s − 3·9-s + 2·13-s + 6·17-s + 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s + 6·45-s − 8·47-s + 49-s − 6·53-s − 6·61-s − 3·63-s − 4·65-s + 4·67-s − 8·71-s − 10·73-s − 16·79-s + 9·81-s + 8·83-s − 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 9-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.768·61-s − 0.377·63-s − 0.496·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s + 0.878·83-s − 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54208\)    =    \(2^{6} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(432.853\)
Root analytic conductor: \(20.8051\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{54208} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 54208,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−14.71375057220915, −14.10729745508485, −13.79497135844297, −13.32357522067834, −12.39420555773819, −11.96522450866148, −11.69546802423313, −11.31778088424449, −10.65379118475940, −9.942359592301097, −9.667960227527052, −8.829144813623910, −8.225405154199764, −8.019090886508062, −7.515484263120878, −6.805496949389982, −6.109346998327415, −5.587978059421554, −5.036264292974459, −4.442927040450731, −3.669020259482086, −3.033905971267796, −2.859476918543032, −1.477338534782348, −1.008754780613775, 0, 1.008754780613775, 1.477338534782348, 2.859476918543032, 3.033905971267796, 3.669020259482086, 4.442927040450731, 5.036264292974459, 5.587978059421554, 6.109346998327415, 6.805496949389982, 7.515484263120878, 8.019090886508062, 8.225405154199764, 8.829144813623910, 9.667960227527052, 9.942359592301097, 10.65379118475940, 11.31778088424449, 11.69546802423313, 11.96522450866148, 12.39420555773819, 13.32357522067834, 13.79497135844297, 14.10729745508485, 14.71375057220915

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