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Properties

Label 2-4000-25.8-c0-0-1
Degree $2$
Conductor $4000$
Sign $0.797 + 0.603i$
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  + (−0.951 − 0.309i)9-s + (0.278 − 0.142i)13-s + (1.76 − 0.278i)17-s + (−1.11 − 1.53i)29-s + (0.412 + 0.809i)37-s + (0.363 − 1.11i)41-s + i·49-s + (1.95 + 0.309i)53-s + (−0.363 − 1.11i)61-s + (0.896 − 1.76i)73-s + (0.809 + 0.587i)81-s + (1.80 − 0.587i)89-s + (0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)9-s + (0.278 − 0.142i)13-s + (1.76 − 0.278i)17-s + (−1.11 − 1.53i)29-s + (0.412 + 0.809i)37-s + (0.363 − 1.11i)41-s + i·49-s + (1.95 + 0.309i)53-s + (−0.363 − 1.11i)61-s + (0.896 − 1.76i)73-s + (0.809 + 0.587i)81-s + (1.80 − 0.587i)89-s + (0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.797 + 0.603i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.797 + 0.603i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.219460150\)
\(L(\frac12)\) \(\approx\) \(1.219460150\)
\(L(1)\) 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 \( 1 \)
good3 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.412 - 0.809i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-1.95 - 0.309i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
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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

−8.478885472324683250174087789972, −7.84057033884969017682353192431, −7.24261362683337572403111034875, −6.02969586048932757912666489954, −5.81571222009704030093654407645, −4.88313122617080892124306144471, −3.79140280183946299740506364236, −3.16506333841467406646810723115, −2.18089425945419952803428963485, −0.796156898929570856522469151921, 1.17462569582701276217779496759, 2.37278246452391450312378259737, 3.31183418884635302665518492425, 3.97119494904179707700201623494, 5.31388922453337902750425552965, 5.50354317954882683202894764082, 6.45531679637615261693075260807, 7.36794006423324002193215077861, 7.964543406605789324298236769241, 8.685562029224743100475070326393

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