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Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 4·11-s − 2·13-s − 3·17-s + 3·23-s + 4·27-s − 6·29-s + 9·31-s + 8·33-s + 4·39-s − 5·41-s − 6·43-s − 9·47-s + 6·51-s + 6·53-s + 8·59-s − 8·61-s + 14·67-s − 6·69-s − 11·71-s − 2·73-s − 9·79-s − 11·81-s + 6·83-s + 12·87-s − 11·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.727·17-s + 0.625·23-s + 0.769·27-s − 1.11·29-s + 1.61·31-s + 1.39·33-s + 0.640·39-s − 0.780·41-s − 0.914·43-s − 1.31·47-s + 0.840·51-s + 0.824·53-s + 1.04·59-s − 1.02·61-s + 1.71·67-s − 0.722·69-s − 1.30·71-s − 0.234·73-s − 1.01·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s − 1.16·89-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: \(0\)
Selberg data: \((2,\ 19600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3720937748\)
\(L(\frac12)\) \(\approx\) \(0.3720937748\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−15.85342161150405, −15.02766208305845, −14.94684700713193, −13.90451694595832, −13.37096521746111, −12.94151951409392, −12.34358709918956, −11.72156649359309, −11.30298276680243, −10.82411712655484, −10.10479010201999, −9.881472813651083, −8.884007540854743, −8.337026289940157, −7.714414285100769, −6.918757805763658, −6.563935364020185, −5.766595872670092, −5.158396013991118, −4.877156416186636, −4.046930992973214, −3.000928913055222, −2.453460458427765, −1.400775936166280, −0.2743213120407153, 0.2743213120407153, 1.400775936166280, 2.453460458427765, 3.000928913055222, 4.046930992973214, 4.877156416186636, 5.158396013991118, 5.766595872670092, 6.563935364020185, 6.918757805763658, 7.714414285100769, 8.337026289940157, 8.884007540854743, 9.881472813651083, 10.10479010201999, 10.82411712655484, 11.30298276680243, 11.72156649359309, 12.34358709918956, 12.94151951409392, 13.37096521746111, 13.90451694595832, 14.94684700713193, 15.02766208305845, 15.85342161150405

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