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Properties

Label 2-140e2-1.1-c1-0-69
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s + 5·11-s − 3·13-s − 17-s + 6·19-s + 6·23-s − 9·27-s − 9·29-s − 4·31-s − 15·33-s − 2·37-s + 9·39-s + 4·41-s + 10·43-s + 47-s + 3·51-s − 4·53-s − 18·57-s − 8·59-s + 8·61-s + 12·67-s − 18·69-s − 8·71-s + 2·73-s − 13·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 1.50·11-s − 0.832·13-s − 0.242·17-s + 1.37·19-s + 1.25·23-s − 1.73·27-s − 1.67·29-s − 0.718·31-s − 2.61·33-s − 0.328·37-s + 1.44·39-s + 0.624·41-s + 1.52·43-s + 0.145·47-s + 0.420·51-s − 0.549·53-s − 2.38·57-s − 1.04·59-s + 1.02·61-s + 1.46·67-s − 2.16·69-s − 0.949·71-s + 0.234·73-s − 1.46·79-s + 81-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: \(1\)
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 \( 1 + p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−16.20082598995448, −15.56693722144207, −14.87254714402490, −14.43027908066322, −13.76367690142359, −12.94469194077055, −12.54771827305510, −12.03881312226215, −11.49049648666276, −11.11573229608996, −10.70637425797649, −9.726534149370717, −9.446686492222880, −8.944473502819573, −7.731294675308703, −7.186643439219735, −6.841471077633745, −6.141376090904421, −5.399152931870650, −5.222292451777762, −4.254468222027174, −3.832159835916044, −2.763906675191133, −1.571203306647625, −1.018415241290297, 0, 1.018415241290297, 1.571203306647625, 2.763906675191133, 3.832159835916044, 4.254468222027174, 5.222292451777762, 5.399152931870650, 6.141376090904421, 6.841471077633745, 7.186643439219735, 7.731294675308703, 8.944473502819573, 9.446686492222880, 9.726534149370717, 10.70637425797649, 11.11573229608996, 11.49049648666276, 12.03881312226215, 12.54771827305510, 12.94469194077055, 13.76367690142359, 14.43027908066322, 14.87254714402490, 15.56693722144207, 16.20082598995448

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