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Properties

Label 2-4000-125.72-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.999 + 0.0439i$
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  + (0.982 + 0.187i)5-s + (−0.368 − 0.929i)9-s + (0.269 − 0.621i)13-s + (0.512 + 1.76i)17-s + (0.929 + 0.368i)25-s + (−0.211 + 1.67i)29-s + (1.46 − 0.327i)37-s + (−0.0604 − 0.961i)41-s + (−0.187 − 0.982i)45-s + (0.951 − 0.309i)49-s + (−0.415 − 1.15i)53-s + (0.123 − 1.96i)61-s + (0.380 − 0.560i)65-s + (0.00591 + 0.0625i)73-s + (−0.728 + 0.684i)81-s + ⋯
L(s)  = 1  + (0.982 + 0.187i)5-s + (−0.368 − 0.929i)9-s + (0.269 − 0.621i)13-s + (0.512 + 1.76i)17-s + (0.929 + 0.368i)25-s + (−0.211 + 1.67i)29-s + (1.46 − 0.327i)37-s + (−0.0604 − 0.961i)41-s + (−0.187 − 0.982i)45-s + (0.951 − 0.309i)49-s + (−0.415 − 1.15i)53-s + (0.123 − 1.96i)61-s + (0.380 − 0.560i)65-s + (0.00591 + 0.0625i)73-s + (−0.728 + 0.684i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.999 + 0.0439i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.999 + 0.0439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.591781927\)
\(L(\frac12)\) \(\approx\) \(1.591781927\)
\(L(1)\) 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 \( 1 + (-0.982 - 0.187i)T \)
good3 \( 1 + (0.368 + 0.929i)T^{2} \)
7 \( 1 + (-0.951 + 0.309i)T^{2} \)
11 \( 1 + (0.876 - 0.481i)T^{2} \)
13 \( 1 + (-0.269 + 0.621i)T + (-0.684 - 0.728i)T^{2} \)
17 \( 1 + (-0.512 - 1.76i)T + (-0.844 + 0.535i)T^{2} \)
19 \( 1 + (0.929 + 0.368i)T^{2} \)
23 \( 1 + (-0.125 + 0.992i)T^{2} \)
29 \( 1 + (0.211 - 1.67i)T + (-0.968 - 0.248i)T^{2} \)
31 \( 1 + (0.535 + 0.844i)T^{2} \)
37 \( 1 + (-1.46 + 0.327i)T + (0.904 - 0.425i)T^{2} \)
41 \( 1 + (0.0604 + 0.961i)T + (-0.992 + 0.125i)T^{2} \)
43 \( 1 + (0.587 + 0.809i)T^{2} \)
47 \( 1 + (-0.998 + 0.0627i)T^{2} \)
53 \( 1 + (0.415 + 1.15i)T + (-0.770 + 0.637i)T^{2} \)
59 \( 1 + (0.187 - 0.982i)T^{2} \)
61 \( 1 + (-0.123 + 1.96i)T + (-0.992 - 0.125i)T^{2} \)
67 \( 1 + (0.248 + 0.968i)T^{2} \)
71 \( 1 + (0.0627 + 0.998i)T^{2} \)
73 \( 1 + (-0.00591 - 0.0625i)T + (-0.982 + 0.187i)T^{2} \)
79 \( 1 + (0.929 - 0.368i)T^{2} \)
83 \( 1 + (-0.368 + 0.929i)T^{2} \)
89 \( 1 + (-0.383 - 0.317i)T + (0.187 + 0.982i)T^{2} \)
97 \( 1 + (-1.22 - 1.57i)T + (-0.248 + 0.968i)T^{2} \)
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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

−8.674076379838413888234063108174, −8.020251031123694894385806411703, −7.00150292502081767767063339472, −6.28366960700105717158218867216, −5.78653233131561130582849873400, −5.09619308609667711860719551638, −3.80216832758435630487865235368, −3.26736943648889931622572518899, −2.15338950915754692406096990922, −1.14385158479943065221687485645, 1.14943795350724557058164068012, 2.37011926819951234405422852834, 2.82986950894615210080691341836, 4.28889178447834729940748101296, 4.90028239739362848374872870326, 5.73215504762658198905712332744, 6.24100811831914710321188064611, 7.27355801084368150322768082439, 7.83253220709052447117389577599, 8.737270327424734751957892682768

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