↑

Properties

Label 2-4000-125.62-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.952 + 0.303i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.844 − 0.535i)5-s + (0.904 − 0.425i)9-s + (1.61 + 0.583i)13-s + (−0.508 + 0.448i)17-s + (0.425 − 0.904i)25-s + (−0.0922 − 0.233i)29-s + (−0.888 − 0.689i)37-s + (−0.374 + 1.96i)41-s + (0.535 − 0.844i)45-s + (0.587 + 0.809i)49-s + (−1.57 + 0.934i)53-s + (−0.316 − 1.65i)61-s + (1.68 − 0.375i)65-s + (0.0525 − 0.180i)73-s + (0.637 − 0.770i)81-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)5-s + (0.904 − 0.425i)9-s + (1.61 + 0.583i)13-s + (−0.508 + 0.448i)17-s + (0.425 − 0.904i)25-s + (−0.0922 − 0.233i)29-s + (−0.888 − 0.689i)37-s + (−0.374 + 1.96i)41-s + (0.535 − 0.844i)45-s + (0.587 + 0.809i)49-s + (−1.57 + 0.934i)53-s + (−0.316 − 1.65i)61-s + (1.68 − 0.375i)65-s + (0.0525 − 0.180i)73-s + (0.637 − 0.770i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.952 + 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.748782484\)
\(L(\frac12)\) \(\approx\) \(1.748782484\)
\(L(1)\) 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 \( 1 + (-0.844 + 0.535i)T \)
good3 \( 1 + (-0.904 + 0.425i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
11 \( 1 + (0.0627 + 0.998i)T^{2} \)
13 \( 1 + (-1.61 - 0.583i)T + (0.770 + 0.637i)T^{2} \)
17 \( 1 + (0.508 - 0.448i)T + (0.125 - 0.992i)T^{2} \)
19 \( 1 + (0.425 - 0.904i)T^{2} \)
23 \( 1 + (0.368 + 0.929i)T^{2} \)
29 \( 1 + (0.0922 + 0.233i)T + (-0.728 + 0.684i)T^{2} \)
31 \( 1 + (-0.992 - 0.125i)T^{2} \)
37 \( 1 + (0.888 + 0.689i)T + (0.248 + 0.968i)T^{2} \)
41 \( 1 + (0.374 - 1.96i)T + (-0.929 - 0.368i)T^{2} \)
43 \( 1 + (-0.951 - 0.309i)T^{2} \)
47 \( 1 + (-0.982 - 0.187i)T^{2} \)
53 \( 1 + (1.57 - 0.934i)T + (0.481 - 0.876i)T^{2} \)
59 \( 1 + (-0.535 - 0.844i)T^{2} \)
61 \( 1 + (0.316 + 1.65i)T + (-0.929 + 0.368i)T^{2} \)
67 \( 1 + (-0.684 + 0.728i)T^{2} \)
71 \( 1 + (-0.187 + 0.982i)T^{2} \)
73 \( 1 + (-0.0525 + 0.180i)T + (-0.844 - 0.535i)T^{2} \)
79 \( 1 + (0.425 + 0.904i)T^{2} \)
83 \( 1 + (0.904 + 0.425i)T^{2} \)
89 \( 1 + (-0.659 - 1.19i)T + (-0.535 + 0.844i)T^{2} \)
97 \( 1 + (1.76 + 0.762i)T + (0.684 + 0.728i)T^{2} \)
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

−8.634293715507469078063746160529, −8.041026940721250649368381271003, −6.92073345008693479245370914738, −6.32966416857529863772848879821, −5.81620498296082437896361850116, −4.70645354044113553657458007800, −4.14223275897281283185259592197, −3.19597600901746338954193460652, −1.84791688929848548419662631762, −1.26475096644631201475832324761, 1.31891000880287242065773106605, 2.16230895765851685884961892635, 3.23629741269720429392511791247, 3.98524608629201810025334925788, 5.07765744305992352976838886239, 5.68290895843690646526784597978, 6.57924332092103992173991348996, 7.01581462768496327614249869492, 7.927113915868654820189169318678, 8.744276953220212164624142499455

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