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Properties

Label 2-4000-125.103-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.0439 - 0.999i$
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  + (0.904 − 0.425i)5-s + (−0.770 + 0.637i)9-s + (−0.180 + 1.91i)13-s + (−1.22 + 1.57i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−1.57 + 0.934i)37-s + (−0.233 + 0.0922i)41-s + (−0.425 + 0.904i)45-s + (−0.951 + 0.309i)49-s + (1.91 + 0.557i)53-s + (−1.68 − 0.666i)61-s + (0.650 + 1.80i)65-s + (0.360 − 1.61i)73-s + (0.187 − 0.982i)81-s + ⋯
L(s)  = 1  + (0.904 − 0.425i)5-s + (−0.770 + 0.637i)9-s + (−0.180 + 1.91i)13-s + (−1.22 + 1.57i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−1.57 + 0.934i)37-s + (−0.233 + 0.0922i)41-s + (−0.425 + 0.904i)45-s + (−0.951 + 0.309i)49-s + (1.91 + 0.557i)53-s + (−1.68 − 0.666i)61-s + (0.650 + 1.80i)65-s + (0.360 − 1.61i)73-s + (0.187 − 0.982i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0439 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0439 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.0439 - 0.999i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.0439 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.112608504\)
\(L(\frac12)\) \(\approx\) \(1.112608504\)
\(L(1)\) 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 \( 1 + (-0.904 + 0.425i)T \)
good3 \( 1 + (0.770 - 0.637i)T^{2} \)
7 \( 1 + (0.951 - 0.309i)T^{2} \)
11 \( 1 + (-0.992 - 0.125i)T^{2} \)
13 \( 1 + (0.180 - 1.91i)T + (-0.982 - 0.187i)T^{2} \)
17 \( 1 + (1.22 - 1.57i)T + (-0.248 - 0.968i)T^{2} \)
19 \( 1 + (0.637 - 0.770i)T^{2} \)
23 \( 1 + (-0.684 + 0.728i)T^{2} \)
29 \( 1 + (0.340 - 0.362i)T + (-0.0627 - 0.998i)T^{2} \)
31 \( 1 + (0.968 - 0.248i)T^{2} \)
37 \( 1 + (1.57 - 0.934i)T + (0.481 - 0.876i)T^{2} \)
41 \( 1 + (0.233 - 0.0922i)T + (0.728 - 0.684i)T^{2} \)
43 \( 1 + (-0.587 - 0.809i)T^{2} \)
47 \( 1 + (0.368 + 0.929i)T^{2} \)
53 \( 1 + (-1.91 - 0.557i)T + (0.844 + 0.535i)T^{2} \)
59 \( 1 + (0.425 + 0.904i)T^{2} \)
61 \( 1 + (1.68 + 0.666i)T + (0.728 + 0.684i)T^{2} \)
67 \( 1 + (-0.998 - 0.0627i)T^{2} \)
71 \( 1 + (-0.929 + 0.368i)T^{2} \)
73 \( 1 + (-0.360 + 1.61i)T + (-0.904 - 0.425i)T^{2} \)
79 \( 1 + (0.637 + 0.770i)T^{2} \)
83 \( 1 + (-0.770 - 0.637i)T^{2} \)
89 \( 1 + (-1.68 + 1.06i)T + (0.425 - 0.904i)T^{2} \)
97 \( 1 + (-1.95 + 0.0613i)T + (0.998 - 0.0627i)T^{2} \)
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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

−8.874010955438326469863798047493, −8.337303618025485506285461060496, −7.25309960497273008311741067376, −6.39919206479778828177046974491, −6.03529669272984719113394541920, −4.93470315465420772828649275714, −4.49847630707836871438384014257, −3.38407848570269008447101557545, −2.02817979536375825319690222686, −1.81733992241110653377350178973, 0.56836580544248936518399511761, 2.19155316805167939535886148509, 2.87884832194065599491588422713, 3.58936438692619066186152882130, 4.98606595396468483822252876695, 5.46936325888064102077649216571, 6.18416975240049498803720118342, 6.96701814359599445873322334706, 7.60199337770892121189197065627, 8.670935245356631584137728087745

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