Properties

Label 2-8400-1.1-c1-0-105
Degree $2$
Conductor $8400$
Sign $-1$
Analytic cond. $67.0743$
Root an. cond. $8.18989$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 2·11-s − 0.921·13-s − 1.07·17-s − 3.07·19-s + 21-s + 2.34·23-s + 27-s − 6.68·29-s + 7.75·31-s − 2·33-s − 10.8·37-s − 0.921·39-s + 6.49·41-s − 6.52·43-s + 4.68·47-s + 49-s − 1.07·51-s − 3.75·53-s − 3.07·57-s − 10.5·59-s − 4.15·61-s + 63-s + 4.68·67-s + 2.34·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s − 0.603·11-s − 0.255·13-s − 0.261·17-s − 0.706·19-s + 0.218·21-s + 0.487·23-s + 0.192·27-s − 1.24·29-s + 1.39·31-s − 0.348·33-s − 1.78·37-s − 0.147·39-s + 1.01·41-s − 0.994·43-s + 0.682·47-s + 0.142·49-s − 0.151·51-s − 0.516·53-s − 0.407·57-s − 1.37·59-s − 0.532·61-s + 0.125·63-s + 0.571·67-s + 0.281·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(67.0743\)
Root analytic conductor: \(8.18989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 0.921T + 13T^{2} \)
17 \( 1 + 1.07T + 17T^{2} \)
19 \( 1 + 3.07T + 19T^{2} \)
23 \( 1 - 2.34T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + 6.52T + 43T^{2} \)
47 \( 1 - 4.68T + 47T^{2} \)
53 \( 1 + 3.75T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 - 4.68T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 7.07T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 - 8.43T + 97T^{2} \)
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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

−7.57389322520809379025531614007, −6.87074677222050492032592965002, −6.12771485872773155172176225113, −5.23331604848420624072848315191, −4.65095523135277921817141685338, −3.86511510760821681442361949879, −2.99663493002885505476199542971, −2.27393248189696305289675508042, −1.42653788348222441197506851037, 0, 1.42653788348222441197506851037, 2.27393248189696305289675508042, 2.99663493002885505476199542971, 3.86511510760821681442361949879, 4.65095523135277921817141685338, 5.23331604848420624072848315191, 6.12771485872773155172176225113, 6.87074677222050492032592965002, 7.57389322520809379025531614007

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