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Properties

Label 2-4000-25.17-c0-0-2
Degree $2$
Conductor $4000$
Sign $0.443 + 0.896i$
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 + (−0.896 − 1.76i)13-s + (−0.142 − 0.896i)17-s + (−1.11 − 1.53i)29-s + (1.58 − 0.809i)37-s + (−0.363 + 1.11i)41-s − i·49-s + (0.0489 − 0.309i)53-s + (0.363 + 1.11i)61-s + (−0.278 − 0.142i)73-s + (0.809 + 0.587i)81-s + (1.80 − 0.587i)89-s + (−1.76 − 0.278i)97-s − 0.618·101-s + (1.53 + 0.5i)109-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)9-s + (−0.896 − 1.76i)13-s + (−0.142 − 0.896i)17-s + (−1.11 − 1.53i)29-s + (1.58 − 0.809i)37-s + (−0.363 + 1.11i)41-s − i·49-s + (0.0489 − 0.309i)53-s + (0.363 + 1.11i)61-s + (−0.278 − 0.142i)73-s + (0.809 + 0.587i)81-s + (1.80 − 0.587i)89-s + (−1.76 − 0.278i)97-s − 0.618·101-s + (1.53 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.443 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215523878\)
\(L(\frac12)\) \(\approx\) \(1.215523878\)
\(L(1)\) 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 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \)
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 \( 1 + (-1.58 + 0.809i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (-0.0489 + 0.309i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \)
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 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
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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

−8.325951245511317579203344336193, −7.59578698862418475996734537593, −7.34582093932608177743243266760, −6.24415856981410485079688219314, −5.45352889028112300227393529523, −4.78257352487981085102458327090, −3.96557487189154611714231957546, −2.90822105014798316506708198786, −2.14485206147288123826099364059, −0.69952285132200704004475906738, 1.45272634367953630068183288294, 2.19462611321061363461237086159, 3.48444567854195544248087335568, 4.26487735037245109466832037990, 4.81898593312863211863415280434, 5.88875441566962487467539298371, 6.74219191672740514947373011864, 7.13967258695315259701561422407, 7.946992168947349799146559467082, 8.936109210704332995079557041838

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