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Properties

Label 2-140e2-1.1-c1-0-102
Degree $2$
Conductor $19600$
Sign $1$
Analytic cond. $156.506$
Root an. cond. $12.5102$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 3·11-s − 5·13-s − 6·17-s + 19-s + 3·23-s + 4·27-s − 6·29-s + 4·31-s + 6·33-s − 11·37-s + 10·39-s + 3·41-s − 10·43-s + 3·47-s + 12·51-s − 3·53-s − 2·57-s − 4·61-s − 4·67-s − 6·69-s − 12·71-s + 4·73-s + 10·79-s − 11·81-s − 12·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.904·11-s − 1.38·13-s − 1.45·17-s + 0.229·19-s + 0.625·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 1.04·33-s − 1.80·37-s + 1.60·39-s + 0.468·41-s − 1.52·43-s + 0.437·47-s + 1.68·51-s − 0.412·53-s − 0.264·57-s − 0.512·61-s − 0.488·67-s − 0.722·69-s − 1.42·71-s + 0.468·73-s + 1.12·79-s − 1.22·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(156.506\)
Root analytic conductor: \(12.5102\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 19600,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.31447028886438, −15.72467624010475, −15.20635532680149, −14.79204192289731, −13.94604424525803, −13.41408916246425, −12.87993725255524, −12.28682287710811, −11.82984405186860, −11.30731035774524, −10.62099796763905, −10.43950985477718, −9.593378966877386, −9.044704004534090, −8.325166952016229, −7.630079442485081, −6.931281174874552, −6.630429844834309, −5.728803054385514, −5.150674354017785, −4.870137471995982, −4.080887599569857, −2.990981017356400, −2.420595793613424, −1.466367647013218, 0, 0, 1.466367647013218, 2.420595793613424, 2.990981017356400, 4.080887599569857, 4.870137471995982, 5.150674354017785, 5.728803054385514, 6.630429844834309, 6.931281174874552, 7.630079442485081, 8.325166952016229, 9.044704004534090, 9.593378966877386, 10.43950985477718, 10.62099796763905, 11.30731035774524, 11.82984405186860, 12.28682287710811, 12.87993725255524, 13.41408916246425, 13.94604424525803, 14.79204192289731, 15.20635532680149, 15.72467624010475, 16.31447028886438

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