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Properties

Label 2-4000-125.98-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.582 + 0.812i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.770 − 0.637i)5-s + (−0.982 + 0.187i)9-s + (0.245 − 0.360i)13-s + (1.42 − 0.842i)17-s + (0.187 − 0.982i)25-s + (−0.961 − 0.0604i)29-s + (0.0175 − 0.0603i)37-s + (0.362 + 0.340i)41-s + (−0.637 + 0.770i)45-s + (0.587 − 0.809i)49-s + (0.288 − 1.29i)53-s + (1.12 − 1.05i)61-s + (−0.0410 − 0.434i)65-s + (−0.621 + 1.72i)73-s + (0.929 − 0.368i)81-s + ⋯
L(s)  = 1  + (0.770 − 0.637i)5-s + (−0.982 + 0.187i)9-s + (0.245 − 0.360i)13-s + (1.42 − 0.842i)17-s + (0.187 − 0.982i)25-s + (−0.961 − 0.0604i)29-s + (0.0175 − 0.0603i)37-s + (0.362 + 0.340i)41-s + (−0.637 + 0.770i)45-s + (0.587 − 0.809i)49-s + (0.288 − 1.29i)53-s + (1.12 − 1.05i)61-s + (−0.0410 − 0.434i)65-s + (−0.621 + 1.72i)73-s + (0.929 − 0.368i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.582 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407100426\)
\(L(\frac12)\) \(\approx\) \(1.407100426\)
\(L(1)\) 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 \( 1 + (-0.770 + 0.637i)T \)
good3 \( 1 + (0.982 - 0.187i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
11 \( 1 + (0.968 - 0.248i)T^{2} \)
13 \( 1 + (-0.245 + 0.360i)T + (-0.368 - 0.929i)T^{2} \)
17 \( 1 + (-1.42 + 0.842i)T + (0.481 - 0.876i)T^{2} \)
19 \( 1 + (0.187 - 0.982i)T^{2} \)
23 \( 1 + (0.998 + 0.0627i)T^{2} \)
29 \( 1 + (0.961 + 0.0604i)T + (0.992 + 0.125i)T^{2} \)
31 \( 1 + (0.876 + 0.481i)T^{2} \)
37 \( 1 + (-0.0175 + 0.0603i)T + (-0.844 - 0.535i)T^{2} \)
41 \( 1 + (-0.362 - 0.340i)T + (0.0627 + 0.998i)T^{2} \)
43 \( 1 + (-0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.684 - 0.728i)T^{2} \)
53 \( 1 + (-0.288 + 1.29i)T + (-0.904 - 0.425i)T^{2} \)
59 \( 1 + (0.637 + 0.770i)T^{2} \)
61 \( 1 + (-1.12 + 1.05i)T + (0.0627 - 0.998i)T^{2} \)
67 \( 1 + (0.125 + 0.992i)T^{2} \)
71 \( 1 + (0.728 + 0.684i)T^{2} \)
73 \( 1 + (0.621 - 1.72i)T + (-0.770 - 0.637i)T^{2} \)
79 \( 1 + (0.187 + 0.982i)T^{2} \)
83 \( 1 + (-0.982 - 0.187i)T^{2} \)
89 \( 1 + (-0.226 + 0.106i)T + (0.637 - 0.770i)T^{2} \)
97 \( 1 + (-1.24 - 1.41i)T + (-0.125 + 0.992i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.474856784617029059999684466129, −7.968780688664796667810087495632, −7.06695682920848949207589300151, −6.10665747514458259897609879781, −5.41875155405395749456480423767, −5.12532529734827911874176045139, −3.85713005085998076902181338488, −2.96441586697343057749602187735, −2.07475412056706664390752638820, −0.846091326562409724932860429908, 1.36719419688712354284960110862, 2.42864684212833163779930150377, 3.24401606012615181768652259894, 3.99713553691727570367242990152, 5.29100431967538723515627111299, 5.84811195801402687008396596896, 6.33089047225590656743176989859, 7.32446077147451859862841431807, 7.927690483203416434619939715818, 8.916423807373751093241132819093

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