Properties

Label 2-1840-1.1-c1-0-1
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.79·3-s − 5-s − 2.79·7-s + 0.208·9-s − 3.79·11-s + 1.20·13-s + 1.79·15-s − 3.79·17-s − 1.20·19-s + 5·21-s − 23-s + 25-s + 5.00·27-s − 1.58·29-s − 10.3·31-s + 6.79·33-s + 2.79·35-s − 4·37-s − 2.16·39-s − 2.20·41-s + 7.16·43-s − 0.208·45-s + 13.5·47-s + 0.791·49-s + 6.79·51-s + 6·53-s + 3.79·55-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.447·5-s − 1.05·7-s + 0.0695·9-s − 1.14·11-s + 0.335·13-s + 0.462·15-s − 0.919·17-s − 0.277·19-s + 1.09·21-s − 0.208·23-s + 0.200·25-s + 0.962·27-s − 0.293·29-s − 1.86·31-s + 1.18·33-s + 0.471·35-s − 0.657·37-s − 0.346·39-s − 0.344·41-s + 1.09·43-s − 0.0311·45-s + 1.98·47-s + 0.113·49-s + 0.950·51-s + 0.824·53-s + 0.511·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3873942648\)
\(L(\frac12)\) \(\approx\) \(0.3873942648\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 + 1.79T + 3T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 - 1.20T + 13T^{2} \)
17 \( 1 + 3.79T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
29 \( 1 + 1.58T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 - 7.16T + 67T^{2} \)
71 \( 1 - 5.37T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 3.16T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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

−9.192558564449213104883704394554, −8.572613416526234713200387429236, −7.45851576494995521440588353022, −6.84408143774689620859325903109, −5.91050763643489754797739481778, −5.43262524105306635957983829240, −4.35724510353962739433974942985, −3.40263296772532377232184229017, −2.31289659133432238693076633442, −0.41599783352405562787186114485, 0.41599783352405562787186114485, 2.31289659133432238693076633442, 3.40263296772532377232184229017, 4.35724510353962739433974942985, 5.43262524105306635957983829240, 5.91050763643489754797739481778, 6.84408143774689620859325903109, 7.45851576494995521440588353022, 8.572613416526234713200387429236, 9.192558564449213104883704394554

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