Properties

Label 2-4200-1.1-c1-0-20
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 2·11-s + 2·13-s − 6·19-s − 21-s + 27-s − 6·29-s + 10·31-s + 2·33-s + 2·39-s + 6·41-s + 8·43-s + 12·47-s + 49-s + 6·53-s − 6·57-s − 6·61-s − 63-s − 4·67-s + 6·71-s + 14·73-s − 2·77-s + 4·79-s + 81-s − 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.37·19-s − 0.218·21-s + 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.348·33-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.794·57-s − 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.712·71-s + 1.63·73-s − 0.227·77-s + 0.450·79-s + 1/9·81-s − 0.643·87-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: $\chi_{4200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398326925\)
\(L(\frac12)\) \(\approx\) \(2.398326925\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.434245297150204148415288146074, −7.78639681133884465502736337734, −6.90387323042488557603119576349, −6.29327202941781976106993694381, −5.57555468896868896251857845353, −4.27630914070959476325236119988, −3.98209334202682588618811268971, −2.87902326798140581591211883883, −2.08648911117884467860480045356, −0.867994520545175267768641485821, 0.867994520545175267768641485821, 2.08648911117884467860480045356, 2.87902326798140581591211883883, 3.98209334202682588618811268971, 4.27630914070959476325236119988, 5.57555468896868896251857845353, 6.29327202941781976106993694381, 6.90387323042488557603119576349, 7.78639681133884465502736337734, 8.434245297150204148415288146074

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