↑

Properties

Label 2-4000-125.3-c0-0-0
Degree 22
Conductor 40004000
Sign 0.963−0.266i0.963 - 0.266i
Analytic cond. 1.996261.99626
Root an. cond. 1.412891.41289
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.684 + 0.728i)5-s + (0.998 − 0.0627i)9-s + (0.743 − 0.843i)13-s + (1.41 + 0.508i)17-s + (−0.0627 + 0.998i)25-s + (−0.742 − 1.35i)29-s + (−1.01 − 0.0958i)37-s + (−1.75 − 0.450i)41-s + (0.728 + 0.684i)45-s + (−0.951 + 0.309i)49-s + (0.313 − 0.461i)53-s + (1.32 − 0.340i)61-s + (1.12 − 0.0353i)65-s + (0.627 + 1.45i)73-s + (0.992 − 0.125i)81-s + ⋯
L(s)  = 1  + (0.684 + 0.728i)5-s + (0.998 − 0.0627i)9-s + (0.743 − 0.843i)13-s + (1.41 + 0.508i)17-s + (−0.0627 + 0.998i)25-s + (−0.742 − 1.35i)29-s + (−1.01 − 0.0958i)37-s + (−1.75 − 0.450i)41-s + (0.728 + 0.684i)45-s + (−0.951 + 0.309i)49-s + (0.313 − 0.461i)53-s + (1.32 − 0.340i)61-s + (1.12 − 0.0353i)65-s + (0.627 + 1.45i)73-s + (0.992 − 0.125i)81-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((0.963−0.266i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((0.963−0.266i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.963−0.266i0.963 - 0.266i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(2753,⋅)\chi_{4000} (2753, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), 0.963−0.266i)(2,\ 4000,\ (\ :0),\ 0.963 - 0.266i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7149438461.714943846
L(12)L(\frac12) ≈\approx 1.7149438461.714943846
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(−0.684−0.728i)T 1 + (-0.684 - 0.728i)T
good3 1+(−0.998+0.0627i)T2 1 + (-0.998 + 0.0627i)T^{2}
7 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
11 1+(−0.425+0.904i)T2 1 + (-0.425 + 0.904i)T^{2}
13 1+(−0.743+0.843i)T+(−0.125−0.992i)T2 1 + (-0.743 + 0.843i)T + (-0.125 - 0.992i)T^{2}
17 1+(−1.41−0.508i)T+(0.770+0.637i)T2 1 + (-1.41 - 0.508i)T + (0.770 + 0.637i)T^{2}
19 1+(−0.0627+0.998i)T2 1 + (-0.0627 + 0.998i)T^{2}
23 1+(0.481+0.876i)T2 1 + (0.481 + 0.876i)T^{2}
29 1+(0.742+1.35i)T+(−0.535+0.844i)T2 1 + (0.742 + 1.35i)T + (-0.535 + 0.844i)T^{2}
31 1+(−0.637+0.770i)T2 1 + (-0.637 + 0.770i)T^{2}
37 1+(1.01+0.0958i)T+(0.982+0.187i)T2 1 + (1.01 + 0.0958i)T + (0.982 + 0.187i)T^{2}
41 1+(1.75+0.450i)T+(0.876+0.481i)T2 1 + (1.75 + 0.450i)T + (0.876 + 0.481i)T^{2}
43 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
47 1+(0.248−0.968i)T2 1 + (0.248 - 0.968i)T^{2}
53 1+(−0.313+0.461i)T+(−0.368−0.929i)T2 1 + (-0.313 + 0.461i)T + (-0.368 - 0.929i)T^{2}
59 1+(−0.728+0.684i)T2 1 + (-0.728 + 0.684i)T^{2}
61 1+(−1.32+0.340i)T+(0.876−0.481i)T2 1 + (-1.32 + 0.340i)T + (0.876 - 0.481i)T^{2}
67 1+(0.844−0.535i)T2 1 + (0.844 - 0.535i)T^{2}
71 1+(0.968+0.248i)T2 1 + (0.968 + 0.248i)T^{2}
73 1+(−0.627−1.45i)T+(−0.684+0.728i)T2 1 + (-0.627 - 1.45i)T + (-0.684 + 0.728i)T^{2}
79 1+(−0.0627−0.998i)T2 1 + (-0.0627 - 0.998i)T^{2}
83 1+(0.998+0.0627i)T2 1 + (0.998 + 0.0627i)T^{2}
89 1+(−0.621+1.57i)T+(−0.728−0.684i)T2 1 + (-0.621 + 1.57i)T + (-0.728 - 0.684i)T^{2}
97 1+(0.512−1.76i)T+(−0.844−0.535i)T2 1 + (0.512 - 1.76i)T + (-0.844 - 0.535i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−8.553860130376247507126641745329, −7.889443828499257562153304375671, −7.15881450732609756463210165828, −6.45752506459838043753464141475, −5.72079247636749501219321714678, −5.11752931329105590825945227523, −3.79299318466904046147138019612, −3.38373275366176879959244188837, −2.17023577444957140787254925945, −1.27043454777770090795485033963, 1.28238903988601333664174999939, 1.82116895637473185427874998180, 3.24442687701479249421595802138, 4.03098322699275004701524266712, 5.02130723905017296570899778406, 5.42221403734480733192675525610, 6.49751364057975111085009532544, 7.01583031967968420876403850497, 7.935076902151283594340775454191, 8.674375553753824008011697437692

Graph of the ZZ-function along the critical line