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Properties

Label 2-4000-125.12-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.368 + 0.929i)5-s + (0.684 + 0.728i)9-s + (0.0625 − 1.99i)13-s + (0.436 − 1.95i)17-s + (−0.728 − 0.684i)25-s + (0.450 − 1.75i)29-s + (−0.415 + 1.15i)37-s + (1.67 − 0.211i)41-s + (−0.929 + 0.368i)45-s + (0.587 + 0.809i)49-s + (−1.01 − 0.0958i)53-s + (0.730 + 0.0922i)61-s + (1.82 + 0.790i)65-s + (1.09 + 0.743i)73-s + (−0.0627 + 0.998i)81-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)5-s + (0.684 + 0.728i)9-s + (0.0625 − 1.99i)13-s + (0.436 − 1.95i)17-s + (−0.728 − 0.684i)25-s + (0.450 − 1.75i)29-s + (−0.415 + 1.15i)37-s + (1.67 − 0.211i)41-s + (−0.929 + 0.368i)45-s + (0.587 + 0.809i)49-s + (−1.01 − 0.0958i)53-s + (0.730 + 0.0922i)61-s + (1.82 + 0.790i)65-s + (1.09 + 0.743i)73-s + (−0.0627 + 0.998i)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: $\chi_{4000} (3137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.952 + 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263538094\)
\(L(\frac12)\) \(\approx\) \(1.263538094\)
\(L(1)\) 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 \( 1 + (0.368 - 0.929i)T \)
good3 \( 1 + (-0.684 - 0.728i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
11 \( 1 + (0.535 - 0.844i)T^{2} \)
13 \( 1 + (-0.0625 + 1.99i)T + (-0.998 - 0.0627i)T^{2} \)
17 \( 1 + (-0.436 + 1.95i)T + (-0.904 - 0.425i)T^{2} \)
19 \( 1 + (-0.728 - 0.684i)T^{2} \)
23 \( 1 + (0.248 - 0.968i)T^{2} \)
29 \( 1 + (-0.450 + 1.75i)T + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (-0.425 + 0.904i)T^{2} \)
37 \( 1 + (0.415 - 1.15i)T + (-0.770 - 0.637i)T^{2} \)
41 \( 1 + (-1.67 + 0.211i)T + (0.968 - 0.248i)T^{2} \)
43 \( 1 + (-0.951 - 0.309i)T^{2} \)
47 \( 1 + (-0.125 - 0.992i)T^{2} \)
53 \( 1 + (1.01 + 0.0958i)T + (0.982 + 0.187i)T^{2} \)
59 \( 1 + (0.929 + 0.368i)T^{2} \)
61 \( 1 + (-0.730 - 0.0922i)T + (0.968 + 0.248i)T^{2} \)
67 \( 1 + (0.481 + 0.876i)T^{2} \)
71 \( 1 + (-0.992 + 0.125i)T^{2} \)
73 \( 1 + (-1.09 - 0.743i)T + (0.368 + 0.929i)T^{2} \)
79 \( 1 + (-0.728 + 0.684i)T^{2} \)
83 \( 1 + (0.684 - 0.728i)T^{2} \)
89 \( 1 + (0.946 - 0.180i)T + (0.929 - 0.368i)T^{2} \)
97 \( 1 + (-0.572 - 0.967i)T + (-0.481 + 0.876i)T^{2} \)
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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

−8.272005114861087749822689413226, −7.73715058349503931319974647189, −7.35924646436563927706763419076, −6.45579129942998011000116087412, −5.58537369382427870622976903042, −4.88225082267621453125852479036, −3.96424887586014056779074296543, −2.90820766889567131695260932869, −2.52913584661496341619867754505, −0.818601724168504721975599420571, 1.25805989345304742219997418230, 1.92377096007436023664827466529, 3.60252953057351018949765363478, 4.04808661003823849214308107299, 4.73020122645673834778886581096, 5.74252967444810176864638328502, 6.51525642593992394232029515922, 7.15323870287521804311296255874, 8.004423981187428121434198760722, 8.870294964869396543248969699788

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