Properties

Label 2-4200-1.1-c1-0-32
Degree $2$
Conductor $4200$
Sign $1$
Analytic cond. $33.5371$
Root an. cond. $5.79112$
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 + 4.77·11-s + 5.77·13-s − 3.77·17-s + 21-s + 3·23-s + 27-s + 3·29-s − 3.77·31-s + 4.77·33-s + 0.772·37-s + 5.77·39-s + 5.77·41-s − 4.54·43-s + 49-s − 3.77·51-s − 3.77·53-s + 5.77·59-s − 1.77·61-s + 63-s − 1.22·67-s + 3·69-s + 1.22·71-s + 2·73-s + 4.77·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.43·11-s + 1.60·13-s − 0.914·17-s + 0.218·21-s + 0.625·23-s + 0.192·27-s + 0.557·29-s − 0.677·31-s + 0.830·33-s + 0.126·37-s + 0.924·39-s + 0.901·41-s − 0.692·43-s + 0.142·49-s − 0.528·51-s − 0.518·53-s + 0.751·59-s − 0.226·61-s + 0.125·63-s − 0.150·67-s + 0.361·69-s + 0.145·71-s + 0.234·73-s + 0.543·77-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.083673071\)
\(L(\frac12)\) \(\approx\) \(3.083673071\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 5.77T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 3.77T + 31T^{2} \)
37 \( 1 - 0.772T + 37T^{2} \)
41 \( 1 - 5.77T + 41T^{2} \)
43 \( 1 + 4.54T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 3.77T + 53T^{2} \)
59 \( 1 - 5.77T + 59T^{2} \)
61 \( 1 + 1.77T + 61T^{2} \)
67 \( 1 + 1.22T + 67T^{2} \)
71 \( 1 - 1.22T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 - 0.455T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.564125929222245225438841281859, −7.80778728436110762376553665567, −6.79466853088417731867587328010, −6.43601585520083427473315652409, −5.47200357624678074070071315245, −4.37201146610065101268986073322, −3.88878696228427796038604877424, −3.02299978118055796504840715675, −1.84205877857434540832259520160, −1.07207460207290721352108095179, 1.07207460207290721352108095179, 1.84205877857434540832259520160, 3.02299978118055796504840715675, 3.88878696228427796038604877424, 4.37201146610065101268986073322, 5.47200357624678074070071315245, 6.43601585520083427473315652409, 6.79466853088417731867587328010, 7.80778728436110762376553665567, 8.564125929222245225438841281859

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