Properties

Label 2-8400-1.1-c1-0-43
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 + 3.26·11-s − 0.340·13-s − 5.75·17-s + 6.49·19-s + 21-s − 8.49·23-s + 27-s − 2·29-s + 8.34·31-s + 3.26·33-s + 6.15·37-s − 0.340·39-s + 0.340·41-s − 8.68·43-s + 49-s − 5.75·51-s + 8.34·53-s + 6.49·57-s − 6.83·59-s + 15.3·61-s + 63-s + 14.8·67-s − 8.49·69-s + 15.9·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s + 0.983·11-s − 0.0943·13-s − 1.39·17-s + 1.49·19-s + 0.218·21-s − 1.77·23-s + 0.192·27-s − 0.371·29-s + 1.49·31-s + 0.567·33-s + 1.01·37-s − 0.0544·39-s + 0.0531·41-s − 1.32·43-s + 0.142·49-s − 0.806·51-s + 1.14·53-s + 0.860·57-s − 0.890·59-s + 1.96·61-s + 0.125·63-s + 1.81·67-s − 1.02·69-s + 1.89·71-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

\(L(1)\) \(\approx\) \(2.908946004\)
\(L(\frac12)\) \(\approx\) \(2.908946004\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.26T + 11T^{2} \)
13 \( 1 + 0.340T + 13T^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 - 6.49T + 19T^{2} \)
23 \( 1 + 8.49T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8.34T + 31T^{2} \)
37 \( 1 - 6.15T + 37T^{2} \)
41 \( 1 - 0.340T + 41T^{2} \)
43 \( 1 + 8.68T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.34T + 53T^{2} \)
59 \( 1 + 6.83T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 - 1.50T + 73T^{2} \)
79 \( 1 + 8.68T + 79T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 6.49T + 97T^{2} \)
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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

−8.016766779038470288508058167976, −7.04151646244583255204261642409, −6.58238788480490738632302788537, −5.76122787194553455772750361095, −4.88130399781308794652524966600, −4.15323636676385428652050726610, −3.60662316340075041267615170085, −2.55068664036218339342980537262, −1.87298135954012854135112759891, −0.830801443534088961792479491938, 0.830801443534088961792479491938, 1.87298135954012854135112759891, 2.55068664036218339342980537262, 3.60662316340075041267615170085, 4.15323636676385428652050726610, 4.88130399781308794652524966600, 5.76122787194553455772750361095, 6.58238788480490738632302788537, 7.04151646244583255204261642409, 8.016766779038470288508058167976

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