Properties

Label 2-8400-1.1-c1-0-104
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 + 1.35·13-s − 3.35·17-s − 5.35·19-s + 21-s − 4.96·23-s + 27-s + 7.92·29-s − 4.57·31-s − 2·33-s − 0.775·37-s + 1.35·39-s + 3.73·41-s + 12.6·43-s − 9.92·47-s + 49-s − 3.35·51-s + 8.57·53-s − 5.35·57-s + 8.62·59-s − 8.70·61-s + 63-s − 9.92·67-s − 4.96·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s − 0.603·11-s + 0.374·13-s − 0.812·17-s − 1.22·19-s + 0.218·21-s − 1.03·23-s + 0.192·27-s + 1.47·29-s − 0.821·31-s − 0.348·33-s − 0.127·37-s + 0.216·39-s + 0.583·41-s + 1.92·43-s − 1.44·47-s + 0.142·49-s − 0.469·51-s + 1.17·53-s − 0.708·57-s + 1.12·59-s − 1.11·61-s + 0.125·63-s − 1.21·67-s − 0.597·69-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 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + 5.35T + 19T^{2} \)
23 \( 1 + 4.96T + 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 + 4.57T + 31T^{2} \)
37 \( 1 + 0.775T + 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + 9.92T + 47T^{2} \)
53 \( 1 - 8.57T + 53T^{2} \)
59 \( 1 - 8.62T + 59T^{2} \)
61 \( 1 + 8.70T + 61T^{2} \)
67 \( 1 + 9.92T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 9.35T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 3.22T + 83T^{2} \)
89 \( 1 - 1.03T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−7.55208754314684051056447152719, −6.81872570677881479675696407270, −6.10666355790468542098737697768, −5.39439111564002220496709732544, −4.29132123597451783806593681388, −4.16966700031446815108481058914, −2.89411502326840353820667977345, −2.32050792731263450448867675621, −1.42054532608382839771610746360, 0, 1.42054532608382839771610746360, 2.32050792731263450448867675621, 2.89411502326840353820667977345, 4.16966700031446815108481058914, 4.29132123597451783806593681388, 5.39439111564002220496709732544, 6.10666355790468542098737697768, 6.81872570677881479675696407270, 7.55208754314684051056447152719

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