Properties

Label 2-4200-1.1-c1-0-29
Degree $2$
Conductor $4200$
Sign $1$
Analytic cond. $33.5371$
Root an. cond. $5.79112$
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 + 4·11-s + 2·13-s + 6·17-s + 4·19-s − 21-s − 8·23-s + 27-s − 2·29-s + 4·33-s + 2·37-s + 2·39-s + 10·41-s − 4·43-s + 49-s + 6·51-s − 14·53-s + 4·57-s + 12·59-s − 2·61-s − 63-s + 4·67-s − 8·69-s − 2·73-s − 4·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.328·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 1/7·49-s + 0.840·51-s − 1.92·53-s + 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.125·63-s + 0.488·67-s − 0.963·69-s − 0.234·73-s − 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(33.5371\)
Root analytic conductor: \(5.79112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.733335573\)
\(L(\frac12)\) \(\approx\) \(2.733335573\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.317615485157378322113674267741, −7.77743443446772037132804004351, −7.03010378769408320045408164406, −6.13892763427373525632497380770, −5.64588020517649894267066587102, −4.42606548256111997498525235700, −3.67806476741052289667892271497, −3.14557720534550427141750493474, −1.90409684093051750893728046113, −0.972092590876197074977141520050, 0.972092590876197074977141520050, 1.90409684093051750893728046113, 3.14557720534550427141750493474, 3.67806476741052289667892271497, 4.42606548256111997498525235700, 5.64588020517649894267066587102, 6.13892763427373525632497380770, 7.03010378769408320045408164406, 7.77743443446772037132804004351, 8.317615485157378322113674267741

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