Properties

Label 2-8400-1.1-c1-0-50
Degree $2$
Conductor $8400$
Sign $1$
Analytic cond. $67.0743$
Root an. cond. $8.18989$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 6·11-s − 2·13-s + 4·17-s + 6·19-s − 21-s − 27-s − 2·29-s + 10·31-s − 6·33-s − 4·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s − 4·51-s + 6·53-s − 6·57-s + 8·59-s − 2·61-s + 63-s + 16·67-s − 10·71-s − 6·73-s + 6·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s − 0.218·21-s − 0.192·27-s − 0.371·29-s + 1.79·31-s − 1.04·33-s − 0.657·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.794·57-s + 1.04·59-s − 0.256·61-s + 0.125·63-s + 1.95·67-s − 1.18·71-s − 0.702·73-s + 0.683·77-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: yes
Arithmetic: yes
Character: $\chi_{8400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.322361287\)
\(L(\frac12)\) \(\approx\) \(2.322361287\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.63642208399059987664156552537, −7.08348656460081845011295837696, −6.43826823199024400071515549204, −5.68036975366865001334852324611, −5.10097823060024598246390702687, −4.26107229251121088868386881866, −3.64436103322453345015934958636, −2.67374112901219525123470788727, −1.44940318569212649034179954751, −0.879874916978760848533151865289, 0.879874916978760848533151865289, 1.44940318569212649034179954751, 2.67374112901219525123470788727, 3.64436103322453345015934958636, 4.26107229251121088868386881866, 5.10097823060024598246390702687, 5.68036975366865001334852324611, 6.43826823199024400071515549204, 7.08348656460081845011295837696, 7.63642208399059987664156552537

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