Properties

Label 2-1200-1.1-c3-0-43
Degree $2$
Conductor $1200$
Sign $-1$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 32·7-s + 9·9-s − 36·11-s + 10·13-s + 78·17-s − 140·19-s − 96·21-s − 192·23-s − 27·27-s + 6·29-s + 16·31-s + 108·33-s + 34·37-s − 30·39-s − 390·41-s − 52·43-s + 408·47-s + 681·49-s − 234·51-s + 114·53-s + 420·57-s − 516·59-s − 58·61-s + 288·63-s − 892·67-s + 576·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.72·7-s + 1/3·9-s − 0.986·11-s + 0.213·13-s + 1.11·17-s − 1.69·19-s − 0.997·21-s − 1.74·23-s − 0.192·27-s + 0.0384·29-s + 0.0926·31-s + 0.569·33-s + 0.151·37-s − 0.123·39-s − 1.48·41-s − 0.184·43-s + 1.26·47-s + 1.98·49-s − 0.642·51-s + 0.295·53-s + 0.975·57-s − 1.13·59-s − 0.121·61-s + 0.575·63-s − 1.62·67-s + 1.00·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(70.8022\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
5 \( 1 \)
good7 \( 1 - 32 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 - 10 T + p^{3} T^{2} \)
17 \( 1 - 78 T + p^{3} T^{2} \)
19 \( 1 + 140 T + p^{3} T^{2} \)
23 \( 1 + 192 T + p^{3} T^{2} \)
29 \( 1 - 6 T + p^{3} T^{2} \)
31 \( 1 - 16 T + p^{3} T^{2} \)
37 \( 1 - 34 T + p^{3} T^{2} \)
41 \( 1 + 390 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 - 408 T + p^{3} T^{2} \)
53 \( 1 - 114 T + p^{3} T^{2} \)
59 \( 1 + 516 T + p^{3} T^{2} \)
61 \( 1 + 58 T + p^{3} T^{2} \)
67 \( 1 + 892 T + p^{3} T^{2} \)
71 \( 1 - 120 T + p^{3} T^{2} \)
73 \( 1 - 646 T + p^{3} T^{2} \)
79 \( 1 - 1168 T + p^{3} T^{2} \)
83 \( 1 + 732 T + p^{3} T^{2} \)
89 \( 1 + 1590 T + p^{3} T^{2} \)
97 \( 1 + 2 p T + p^{3} T^{2} \)
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

−8.735229703157067682633599621954, −8.045205815811440097860446477827, −7.56239864962174165790575892292, −6.28499740097241640823993570310, −5.49479052150955976739072968344, −4.76123073111374433511542300960, −3.92748470922967557762775377781, −2.32444286707874308619545067967, −1.43649291851529844981664610270, 0, 1.43649291851529844981664610270, 2.32444286707874308619545067967, 3.92748470922967557762775377781, 4.76123073111374433511542300960, 5.49479052150955976739072968344, 6.28499740097241640823993570310, 7.56239864962174165790575892292, 8.045205815811440097860446477827, 8.735229703157067682633599621954

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