Properties

Label 2-7400-1.1-c1-0-119
Degree $2$
Conductor $7400$
Sign $-1$
Analytic cond. $59.0892$
Root an. cond. $7.68695$
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-s + 2·7-s − 2·9-s − 3·11-s + 7·17-s − 7·19-s − 2·21-s − 6·23-s + 5·27-s + 8·29-s + 6·31-s + 3·33-s + 37-s − 3·41-s + 4·43-s − 4·47-s − 3·49-s − 7·51-s − 6·53-s + 7·57-s + 8·59-s + 6·61-s − 4·63-s − 3·67-s + 6·69-s − 3·73-s − 6·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.904·11-s + 1.69·17-s − 1.60·19-s − 0.436·21-s − 1.25·23-s + 0.962·27-s + 1.48·29-s + 1.07·31-s + 0.522·33-s + 0.164·37-s − 0.468·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.980·51-s − 0.824·53-s + 0.927·57-s + 1.04·59-s + 0.768·61-s − 0.503·63-s − 0.366·67-s + 0.722·69-s − 0.351·73-s − 0.683·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7400\)    =    \(2^{3} \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(59.0892\)
Root analytic conductor: \(7.68695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
37 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.86241272555018566394234134396, −6.65944481551843387504583554349, −6.12701404250427732718845031156, −5.41466199869469157911102670563, −4.86507065237673191631809191096, −4.09909022832155185044512516519, −3.01678569162726702084851569279, −2.30311963367282470982208482537, −1.16521427366890342224294531113, 0, 1.16521427366890342224294531113, 2.30311963367282470982208482537, 3.01678569162726702084851569279, 4.09909022832155185044512516519, 4.86507065237673191631809191096, 5.41466199869469157911102670563, 6.12701404250427732718845031156, 6.65944481551843387504583554349, 7.86241272555018566394234134396

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