Properties

Label 2-8400-1.1-c1-0-42
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 0.387·11-s + 2.96·13-s − 3.35·17-s + 2.96·19-s + 21-s − 0.962·23-s + 27-s + 1.22·29-s − 2.96·31-s − 0.387·33-s − 5.92·37-s + 2.96·39-s + 1.03·41-s + 10.7·43-s + 3.22·47-s + 49-s − 3.35·51-s + 5.66·53-s + 2.96·57-s − 3.22·59-s + 14.6·61-s + 63-s − 0.962·69-s − 5.53·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s − 0.116·11-s + 0.821·13-s − 0.812·17-s + 0.679·19-s + 0.218·21-s − 0.200·23-s + 0.192·27-s + 0.227·29-s − 0.532·31-s − 0.0675·33-s − 0.974·37-s + 0.474·39-s + 0.162·41-s + 1.63·43-s + 0.470·47-s + 0.142·49-s − 0.469·51-s + 0.777·53-s + 0.392·57-s − 0.419·59-s + 1.87·61-s + 0.125·63-s − 0.115·69-s − 0.657·71-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: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.856449800\)
\(L(\frac12)\) \(\approx\) \(2.856449800\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.387T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 - 2.96T + 19T^{2} \)
23 \( 1 + 0.962T + 23T^{2} \)
29 \( 1 - 1.22T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 + 5.92T + 37T^{2} \)
41 \( 1 - 1.03T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 - 5.66T + 53T^{2} \)
59 \( 1 + 3.22T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 5.53T + 71T^{2} \)
73 \( 1 + 6.18T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 3.22T + 83T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 - 7.73T + 97T^{2} \)
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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

−7.75820377281065732626945110220, −7.25907137320619684979395325476, −6.47364368908625904559807727802, −5.70998391123050498846119831687, −4.98314834828739987860795430481, −4.12209253309611255921931639630, −3.56074499534995349318177843388, −2.60290668251388933044722838760, −1.85458639150356205834873124667, −0.822791417782459046296341571423, 0.822791417782459046296341571423, 1.85458639150356205834873124667, 2.60290668251388933044722838760, 3.56074499534995349318177843388, 4.12209253309611255921931639630, 4.98314834828739987860795430481, 5.70998391123050498846119831687, 6.47364368908625904559807727802, 7.25907137320619684979395325476, 7.75820377281065732626945110220

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