Properties

Label 2-4000-125.78-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.622 + 0.782i$
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  + (−0.125 − 0.992i)5-s + (0.248 − 0.968i)9-s + (−0.0319 − 0.0540i)13-s + (−0.967 − 1.42i)17-s + (−0.968 + 0.248i)25-s + (−0.666 − 0.313i)29-s + (−0.775 + 1.79i)37-s + (1.05 − 1.65i)41-s + (−0.992 − 0.125i)45-s + (−0.951 + 0.309i)49-s + (0.0212 + 0.677i)53-s + (−0.134 − 0.211i)61-s + (−0.0496 + 0.0385i)65-s + (0.418 − 0.369i)73-s + (−0.876 − 0.481i)81-s + ⋯
L(s)  = 1  + (−0.125 − 0.992i)5-s + (0.248 − 0.968i)9-s + (−0.0319 − 0.0540i)13-s + (−0.967 − 1.42i)17-s + (−0.968 + 0.248i)25-s + (−0.666 − 0.313i)29-s + (−0.775 + 1.79i)37-s + (1.05 − 1.65i)41-s + (−0.992 − 0.125i)45-s + (−0.951 + 0.309i)49-s + (0.0212 + 0.677i)53-s + (−0.134 − 0.211i)61-s + (−0.0496 + 0.0385i)65-s + (0.418 − 0.369i)73-s + (−0.876 − 0.481i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9479400844\)
\(L(\frac12)\) \(\approx\) \(0.9479400844\)
\(L(1)\) 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 \( 1 + (0.125 + 0.992i)T \)
good3 \( 1 + (-0.248 + 0.968i)T^{2} \)
7 \( 1 + (0.951 - 0.309i)T^{2} \)
11 \( 1 + (-0.187 - 0.982i)T^{2} \)
13 \( 1 + (0.0319 + 0.0540i)T + (-0.481 + 0.876i)T^{2} \)
17 \( 1 + (0.967 + 1.42i)T + (-0.368 + 0.929i)T^{2} \)
19 \( 1 + (-0.968 + 0.248i)T^{2} \)
23 \( 1 + (-0.904 - 0.425i)T^{2} \)
29 \( 1 + (0.666 + 0.313i)T + (0.637 + 0.770i)T^{2} \)
31 \( 1 + (-0.929 - 0.368i)T^{2} \)
37 \( 1 + (0.775 - 1.79i)T + (-0.684 - 0.728i)T^{2} \)
41 \( 1 + (-1.05 + 1.65i)T + (-0.425 - 0.904i)T^{2} \)
43 \( 1 + (-0.587 - 0.809i)T^{2} \)
47 \( 1 + (-0.844 - 0.535i)T^{2} \)
53 \( 1 + (-0.0212 - 0.677i)T + (-0.998 + 0.0627i)T^{2} \)
59 \( 1 + (0.992 - 0.125i)T^{2} \)
61 \( 1 + (0.134 + 0.211i)T + (-0.425 + 0.904i)T^{2} \)
67 \( 1 + (0.770 + 0.637i)T^{2} \)
71 \( 1 + (0.535 - 0.844i)T^{2} \)
73 \( 1 + (-0.418 + 0.369i)T + (0.125 - 0.992i)T^{2} \)
79 \( 1 + (-0.968 - 0.248i)T^{2} \)
83 \( 1 + (0.248 + 0.968i)T^{2} \)
89 \( 1 + (-1.53 - 0.0967i)T + (0.992 + 0.125i)T^{2} \)
97 \( 1 + (0.448 + 1.24i)T + (-0.770 + 0.637i)T^{2} \)
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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

−8.495013985994579358062365632007, −7.62593617837046680925952886273, −6.93209002914580782799497182355, −6.16868227450346861090002534303, −5.27432537112189194829644125516, −4.59819539387075887918963955801, −3.87222519777569142027687464364, −2.88460325475332306984436479463, −1.69715206069098954973677213301, −0.51056899168246572856474907228, 1.78438947436148931837801963950, 2.44282506920790473116604141680, 3.56082626851589877003263050359, 4.22759537988097114593788442185, 5.17551797974435950828463966968, 6.07852124172147611311538305834, 6.67739642322542187144380032479, 7.49258755898252757625399848547, 8.003468656478580529387136164260, 8.814950528955203079432071443413

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