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Properties

Label 2-4000-125.23-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.592 + 0.805i$
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  + (−0.248 + 0.968i)5-s + (−0.481 − 0.876i)9-s + (−1.44 − 0.418i)13-s + (0.221 − 0.512i)17-s + (−0.876 − 0.481i)25-s + (−1.05 − 0.872i)29-s + (−1.88 + 0.0591i)37-s + (−0.313 + 0.666i)41-s + (0.968 − 0.248i)45-s + (0.587 − 0.809i)49-s + (0.141 − 0.124i)53-s + (0.211 + 0.450i)61-s + (0.763 − 1.29i)65-s + (−1.54 − 1.19i)73-s + (−0.535 + 0.844i)81-s + ⋯
L(s)  = 1  + (−0.248 + 0.968i)5-s + (−0.481 − 0.876i)9-s + (−1.44 − 0.418i)13-s + (0.221 − 0.512i)17-s + (−0.876 − 0.481i)25-s + (−1.05 − 0.872i)29-s + (−1.88 + 0.0591i)37-s + (−0.313 + 0.666i)41-s + (0.968 − 0.248i)45-s + (0.587 − 0.809i)49-s + (0.141 − 0.124i)53-s + (0.211 + 0.450i)61-s + (0.763 − 1.29i)65-s + (−1.54 − 1.19i)73-s + (−0.535 + 0.844i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3831120005\)
\(L(\frac12)\) \(\approx\) \(0.3831120005\)
\(L(1)\) 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 \( 1 + (0.248 - 0.968i)T \)
good3 \( 1 + (0.481 + 0.876i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
11 \( 1 + (-0.929 - 0.368i)T^{2} \)
13 \( 1 + (1.44 + 0.418i)T + (0.844 + 0.535i)T^{2} \)
17 \( 1 + (-0.221 + 0.512i)T + (-0.684 - 0.728i)T^{2} \)
19 \( 1 + (-0.876 - 0.481i)T^{2} \)
23 \( 1 + (-0.770 - 0.637i)T^{2} \)
29 \( 1 + (1.05 + 0.872i)T + (0.187 + 0.982i)T^{2} \)
31 \( 1 + (0.728 - 0.684i)T^{2} \)
37 \( 1 + (1.88 - 0.0591i)T + (0.998 - 0.0627i)T^{2} \)
41 \( 1 + (0.313 - 0.666i)T + (-0.637 - 0.770i)T^{2} \)
43 \( 1 + (-0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.904 + 0.425i)T^{2} \)
53 \( 1 + (-0.141 + 0.124i)T + (0.125 - 0.992i)T^{2} \)
59 \( 1 + (-0.968 - 0.248i)T^{2} \)
61 \( 1 + (-0.211 - 0.450i)T + (-0.637 + 0.770i)T^{2} \)
67 \( 1 + (0.982 + 0.187i)T^{2} \)
71 \( 1 + (-0.425 + 0.904i)T^{2} \)
73 \( 1 + (1.54 + 1.19i)T + (0.248 + 0.968i)T^{2} \)
79 \( 1 + (-0.876 + 0.481i)T^{2} \)
83 \( 1 + (-0.481 + 0.876i)T^{2} \)
89 \( 1 + (0.246 + 1.94i)T + (-0.968 + 0.248i)T^{2} \)
97 \( 1 + (0.115 + 1.22i)T + (-0.982 + 0.187i)T^{2} \)
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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

−8.347449813005905438055967721568, −7.32835353390033241472456750986, −7.17499592156101609901919495236, −6.16184404383387332790168948986, −5.50945646339856828764500518598, −4.56363363258262126903271606609, −3.52816896808677500682725405154, −2.96907873766308664704055361062, −2.04136683000627423859640613377, −0.19830076886808140690456727962, 1.57553487714645894974581213402, 2.41429444957277014426289649601, 3.59013698679084444444097952466, 4.46373826020015677873414752997, 5.24986028443529878734039861887, 5.56890785554955119430805416035, 6.87606572841876547253143382561, 7.49441464194237263588618804853, 8.171421682575680209485719975213, 8.865662292011108774092447522215

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