Properties

Label 2-140e2-1.1-c1-0-39
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  + 3-s − 2·9-s + 6·11-s + 4·13-s − 2·19-s − 3·23-s − 5·27-s − 3·29-s − 8·31-s + 6·33-s + 4·37-s + 4·39-s + 9·41-s − 7·43-s + 6·53-s − 2·57-s + 6·59-s + 5·61-s + 5·67-s − 3·69-s + 6·71-s + 16·73-s − 2·79-s + 81-s + 3·83-s − 3·87-s − 15·89-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.80·11-s + 1.10·13-s − 0.458·19-s − 0.625·23-s − 0.962·27-s − 0.557·29-s − 1.43·31-s + 1.04·33-s + 0.657·37-s + 0.640·39-s + 1.40·41-s − 1.06·43-s + 0.824·53-s − 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.610·67-s − 0.361·69-s + 0.712·71-s + 1.87·73-s − 0.225·79-s + 1/9·81-s + 0.329·83-s − 0.321·87-s − 1.58·89-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

\(L(1)\) \(\approx\) \(2.968392334\)
\(L(\frac12)\) \(\approx\) \(2.968392334\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−15.59681280339425, −14.95678966364684, −14.56978592883142, −14.13799107234524, −13.65333446328388, −13.00182966868333, −12.45148906865371, −11.72552991535894, −11.22801132934831, −10.96085723613669, −9.955401869215799, −9.321135338901088, −9.040261790498875, −8.367222141534411, −7.984730370992098, −7.033040900802207, −6.551173848211669, −5.879001526173643, −5.417818174102269, −4.187588071941270, −3.863285683719213, −3.322280838520319, −2.290778595851608, −1.665629830898137, −0.7039459031113941, 0.7039459031113941, 1.665629830898137, 2.290778595851608, 3.322280838520319, 3.863285683719213, 4.187588071941270, 5.417818174102269, 5.879001526173643, 6.551173848211669, 7.033040900802207, 7.984730370992098, 8.367222141534411, 9.040261790498875, 9.321135338901088, 9.955401869215799, 10.96085723613669, 11.22801132934831, 11.72552991535894, 12.45148906865371, 13.00182966868333, 13.65333446328388, 14.13799107234524, 14.56978592883142, 14.95678966364684, 15.59681280339425

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