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Properties

Label 2-4000-125.2-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.303 - 0.952i$
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  + (−0.844 − 0.535i)5-s + (−0.904 − 0.425i)9-s + (−0.344 − 0.957i)13-s + (−1.24 + 1.41i)17-s + (0.425 + 0.904i)25-s + (−0.0922 + 0.233i)29-s + (1.01 + 1.30i)37-s + (0.374 + 1.96i)41-s + (0.535 + 0.844i)45-s + (−0.587 + 0.809i)49-s + (−0.404 + 0.683i)53-s + (0.316 − 1.65i)61-s + (−0.222 + 0.993i)65-s + (−1.91 + 0.555i)73-s + (0.637 + 0.770i)81-s + ⋯
L(s)  = 1  + (−0.844 − 0.535i)5-s + (−0.904 − 0.425i)9-s + (−0.344 − 0.957i)13-s + (−1.24 + 1.41i)17-s + (0.425 + 0.904i)25-s + (−0.0922 + 0.233i)29-s + (1.01 + 1.30i)37-s + (0.374 + 1.96i)41-s + (0.535 + 0.844i)45-s + (−0.587 + 0.809i)49-s + (−0.404 + 0.683i)53-s + (0.316 − 1.65i)61-s + (−0.222 + 0.993i)65-s + (−1.91 + 0.555i)73-s + (0.637 + 0.770i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3751744676\)
\(L(\frac12)\) \(\approx\) \(0.3751744676\)
\(L(1)\) 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 \( 1 + (0.844 + 0.535i)T \)
good3 \( 1 + (0.904 + 0.425i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
11 \( 1 + (0.0627 - 0.998i)T^{2} \)
13 \( 1 + (0.344 + 0.957i)T + (-0.770 + 0.637i)T^{2} \)
17 \( 1 + (1.24 - 1.41i)T + (-0.125 - 0.992i)T^{2} \)
19 \( 1 + (0.425 + 0.904i)T^{2} \)
23 \( 1 + (-0.368 + 0.929i)T^{2} \)
29 \( 1 + (0.0922 - 0.233i)T + (-0.728 - 0.684i)T^{2} \)
31 \( 1 + (-0.992 + 0.125i)T^{2} \)
37 \( 1 + (-1.01 - 1.30i)T + (-0.248 + 0.968i)T^{2} \)
41 \( 1 + (-0.374 - 1.96i)T + (-0.929 + 0.368i)T^{2} \)
43 \( 1 + (0.951 - 0.309i)T^{2} \)
47 \( 1 + (0.982 - 0.187i)T^{2} \)
53 \( 1 + (0.404 - 0.683i)T + (-0.481 - 0.876i)T^{2} \)
59 \( 1 + (-0.535 + 0.844i)T^{2} \)
61 \( 1 + (-0.316 + 1.65i)T + (-0.929 - 0.368i)T^{2} \)
67 \( 1 + (0.684 + 0.728i)T^{2} \)
71 \( 1 + (-0.187 - 0.982i)T^{2} \)
73 \( 1 + (1.91 - 0.555i)T + (0.844 - 0.535i)T^{2} \)
79 \( 1 + (0.425 - 0.904i)T^{2} \)
83 \( 1 + (-0.904 + 0.425i)T^{2} \)
89 \( 1 + (-0.659 + 1.19i)T + (-0.535 - 0.844i)T^{2} \)
97 \( 1 + (0.221 + 0.512i)T + (-0.684 + 0.728i)T^{2} \)
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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

−8.684804917163130566886089935282, −8.142735320545064198340885822909, −7.65334394031779652968438259773, −6.47720940698527892601753440096, −6.02273111608300230098562413926, −4.97419578268359161652012895554, −4.36264361376548981914051923053, −3.42454486767874358746988946629, −2.69099249720673125715962066697, −1.26552875340877029170429066174, 0.21500873562656545217412207866, 2.21034426263210305262649357584, 2.75588940027289519879293057650, 3.87342462417222137202887386289, 4.53837532208357406626792481300, 5.37528767701219904563464732098, 6.31312190435701965654460918397, 7.14150602706473231239388078450, 7.46992177007302234214101629501, 8.495791392465034416927620109737

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