↑

Properties

Label 2-2380-85.84-c1-0-10
Degree $2$
Conductor $2380$
Sign $0.00528 - 0.999i$
Analytic cond. $19.0043$
Root an. cond. $4.35940$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.807·3-s + (−2.19 − 0.442i)5-s + 7-s − 2.34·9-s + 0.748i·11-s − 2.28i·13-s + (−1.76 − 0.356i)15-s + (0.793 + 4.04i)17-s + 0.712·19-s + 0.807·21-s + 2.80·23-s + (4.60 + 1.93i)25-s − 4.31·27-s − 5.30i·29-s + 4.52i·31-s + ⋯
L(s)  = 1  + 0.466·3-s + (−0.980 − 0.197i)5-s + 0.377·7-s − 0.782·9-s + 0.225i·11-s − 0.634i·13-s + (−0.456 − 0.0921i)15-s + (0.192 + 0.981i)17-s + 0.163·19-s + 0.176·21-s + 0.585·23-s + (0.921 + 0.387i)25-s − 0.830·27-s − 0.985i·29-s + 0.813i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00528 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00528 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.00528 - 0.999i$
Analytic conductor: \(19.0043\)
Root analytic conductor: \(4.35940\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :1/2),\ 0.00528 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093518076\)
\(L(\frac12)\) \(\approx\) \(1.093518076\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 0.442i)T \)
7 \( 1 - T \)
17 \( 1 + (-0.793 - 4.04i)T \)
good3 \( 1 - 0.807T + 3T^{2} \)
11 \( 1 - 0.748iT - 11T^{2} \)
13 \( 1 + 2.28iT - 13T^{2} \)
19 \( 1 - 0.712T + 19T^{2} \)
23 \( 1 - 2.80T + 23T^{2} \)
29 \( 1 + 5.30iT - 29T^{2} \)
31 \( 1 - 4.52iT - 31T^{2} \)
37 \( 1 + 9.33T + 37T^{2} \)
41 \( 1 + 1.92iT - 41T^{2} \)
43 \( 1 - 6.53iT - 43T^{2} \)
47 \( 1 - 8.50iT - 47T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 - 6.74T + 59T^{2} \)
61 \( 1 + 1.53iT - 61T^{2} \)
67 \( 1 - 4.60iT - 67T^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 + 3.39T + 73T^{2} \)
79 \( 1 - 5.22iT - 79T^{2} \)
83 \( 1 - 6.69iT - 83T^{2} \)
89 \( 1 - 0.864T + 89T^{2} \)
97 \( 1 - 7.00T + 97T^{2} \)
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

−8.857710935331612747348184215059, −8.391875046426880006215340466582, −7.80245574168903808411590514276, −7.08393391188577328568305782215, −5.97079844542143435448425242222, −5.17869361929197412832555075526, −4.24767789835717660958189674422, −3.42662842252311870172671959057, −2.61399398776762593250018339092, −1.19694942577142775441620141438, 0.37607809783113870232034169810, 2.00858030993483123748476219255, 3.12758167737831603643060002697, 3.68650581546677480523474306497, 4.81467405970176831845347600849, 5.46639454017462374292822606493, 6.73702463494511338874171168517, 7.26908777705087120762042598150, 8.108243900983099919824054172338, 8.719605929627059956152921141728

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