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Properties

Label 2-280e2-1.1-c1-0-117
Degree $2$
Conductor $78400$
Sign $-1$
Analytic cond. $626.027$
Root an. cond. $25.0205$
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  − 3·9-s − 4·11-s − 2·13-s − 6·17-s − 8·19-s − 6·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s + 8·47-s + 6·53-s − 6·61-s + 4·67-s + 8·71-s + 10·73-s − 16·79-s + 9·81-s + 8·83-s + 6·89-s − 6·97-s + 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 0.824·53-s − 0.768·61-s + 0.488·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 81-s + 0.878·83-s + 0.635·89-s − 0.609·97-s + 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(626.027\)
Root analytic conductor: \(25.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
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

−14.18660396350996, −13.74964407076340, −13.26520446650377, −12.80135234647956, −12.37183442060860, −11.74869162294645, −11.11920375978353, −10.87486734743779, −10.33100958161450, −9.853287463104078, −8.953208195885528, −8.802259319767058, −8.230308356263809, −7.693336460620826, −7.120652290652195, −6.420388376262515, −6.079173599050146, −5.384470970043073, −4.854098863404727, −4.332986729468430, −3.689276522224822, −2.807368080200392, −2.328927080596244, −2.026886761995913, −0.6124324117162262, 0, 0.6124324117162262, 2.026886761995913, 2.328927080596244, 2.807368080200392, 3.689276522224822, 4.332986729468430, 4.854098863404727, 5.384470970043073, 6.079173599050146, 6.420388376262515, 7.120652290652195, 7.693336460620826, 8.230308356263809, 8.802259319767058, 8.953208195885528, 9.853287463104078, 10.33100958161450, 10.87486734743779, 11.11920375978353, 11.74869162294645, 12.37183442060860, 12.80135234647956, 13.26520446650377, 13.74964407076340, 14.18660396350996

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