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Properties

Label 2-4000-125.92-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  + (−0.982 + 0.187i)5-s + (0.368 − 0.929i)9-s + (−1.72 + 0.747i)13-s + (0.762 + 0.221i)17-s + (0.929 − 0.368i)25-s + (−0.211 − 1.67i)29-s + (0.288 − 1.29i)37-s + (0.0604 − 0.961i)41-s + (−0.187 + 0.982i)45-s + (−0.951 − 0.309i)49-s + (1.48 + 0.535i)53-s + (−0.123 − 1.96i)61-s + (1.55 − 1.05i)65-s + (−1.99 − 0.188i)73-s + (−0.728 − 0.684i)81-s + ⋯
L(s)  = 1  + (−0.982 + 0.187i)5-s + (0.368 − 0.929i)9-s + (−1.72 + 0.747i)13-s + (0.762 + 0.221i)17-s + (0.929 − 0.368i)25-s + (−0.211 − 1.67i)29-s + (0.288 − 1.29i)37-s + (0.0604 − 0.961i)41-s + (−0.187 + 0.982i)45-s + (−0.951 − 0.309i)49-s + (1.48 + 0.535i)53-s + (−0.123 − 1.96i)61-s + (1.55 − 1.05i)65-s + (−1.99 − 0.188i)73-s + (−0.728 − 0.684i)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: $\chi_{4000} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.0439 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7436209623\)
\(L(\frac12)\) \(\approx\) \(0.7436209623\)
\(L(1)\) 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 \( 1 + (0.982 - 0.187i)T \)
good3 \( 1 + (-0.368 + 0.929i)T^{2} \)
7 \( 1 + (0.951 + 0.309i)T^{2} \)
11 \( 1 + (0.876 + 0.481i)T^{2} \)
13 \( 1 + (1.72 - 0.747i)T + (0.684 - 0.728i)T^{2} \)
17 \( 1 + (-0.762 - 0.221i)T + (0.844 + 0.535i)T^{2} \)
19 \( 1 + (0.929 - 0.368i)T^{2} \)
23 \( 1 + (0.125 + 0.992i)T^{2} \)
29 \( 1 + (0.211 + 1.67i)T + (-0.968 + 0.248i)T^{2} \)
31 \( 1 + (0.535 - 0.844i)T^{2} \)
37 \( 1 + (-0.288 + 1.29i)T + (-0.904 - 0.425i)T^{2} \)
41 \( 1 + (-0.0604 + 0.961i)T + (-0.992 - 0.125i)T^{2} \)
43 \( 1 + (-0.587 + 0.809i)T^{2} \)
47 \( 1 + (0.998 + 0.0627i)T^{2} \)
53 \( 1 + (-1.48 - 0.535i)T + (0.770 + 0.637i)T^{2} \)
59 \( 1 + (0.187 + 0.982i)T^{2} \)
61 \( 1 + (0.123 + 1.96i)T + (-0.992 + 0.125i)T^{2} \)
67 \( 1 + (-0.248 + 0.968i)T^{2} \)
71 \( 1 + (0.0627 - 0.998i)T^{2} \)
73 \( 1 + (1.99 + 0.188i)T + (0.982 + 0.187i)T^{2} \)
79 \( 1 + (0.929 + 0.368i)T^{2} \)
83 \( 1 + (0.368 + 0.929i)T^{2} \)
89 \( 1 + (-0.383 + 0.317i)T + (0.187 - 0.982i)T^{2} \)
97 \( 1 + (0.148 + 0.115i)T + (0.248 + 0.968i)T^{2} \)
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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

−8.364266433948176899841495262617, −7.52081497375506419517497634811, −7.20222367851638464471750555275, −6.37225052838781324295818216937, −5.45240734212841855041984991437, −4.43274770849492378801473877675, −3.97607540372879619300471000542, −3.03758179468651275273186992664, −2.02318463626196211266873095396, −0.44428843303494600532296142044, 1.26624431616060692405307275869, 2.62915564087727394824882157910, 3.28746613487335032458488484582, 4.46741093744834633271496245126, 4.94295902978291554678291990976, 5.61281037277824106363355004946, 6.97489207395066008788791470250, 7.37487723212061969312023402683, 7.978422976325669406529634864617, 8.602468674403917847976613352380

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