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Properties

Label 2-4000-125.77-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.805 + 0.592i$
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  + (0.998 − 0.0627i)5-s + (0.125 − 0.992i)9-s + (−1.45 + 1.12i)13-s + (0.115 − 1.22i)17-s + (0.992 − 0.125i)25-s + (1.65 − 1.05i)29-s + (0.313 − 0.461i)37-s + (1.35 − 0.742i)41-s + (0.0627 − 0.998i)45-s + (−0.587 + 0.809i)49-s + (0.400 − 0.173i)53-s + (1.74 + 0.961i)61-s + (−1.37 + 1.21i)65-s + (0.0540 − 1.72i)73-s + (−0.968 − 0.248i)81-s + ⋯
L(s)  = 1  + (0.998 − 0.0627i)5-s + (0.125 − 0.992i)9-s + (−1.45 + 1.12i)13-s + (0.115 − 1.22i)17-s + (0.992 − 0.125i)25-s + (1.65 − 1.05i)29-s + (0.313 − 0.461i)37-s + (1.35 − 0.742i)41-s + (0.0627 − 0.998i)45-s + (−0.587 + 0.809i)49-s + (0.400 − 0.173i)53-s + (1.74 + 0.961i)61-s + (−1.37 + 1.21i)65-s + (0.0540 − 1.72i)73-s + (−0.968 − 0.248i)81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.510828596\)
\(L(\frac12)\) \(\approx\) \(1.510828596\)
\(L(1)\) 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 \( 1 + (-0.998 + 0.0627i)T \)
good3 \( 1 + (-0.125 + 0.992i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
11 \( 1 + (-0.637 + 0.770i)T^{2} \)
13 \( 1 + (1.45 - 1.12i)T + (0.248 - 0.968i)T^{2} \)
17 \( 1 + (-0.115 + 1.22i)T + (-0.982 - 0.187i)T^{2} \)
19 \( 1 + (0.992 - 0.125i)T^{2} \)
23 \( 1 + (0.844 - 0.535i)T^{2} \)
29 \( 1 + (-1.65 + 1.05i)T + (0.425 - 0.904i)T^{2} \)
31 \( 1 + (-0.187 + 0.982i)T^{2} \)
37 \( 1 + (-0.313 + 0.461i)T + (-0.368 - 0.929i)T^{2} \)
41 \( 1 + (-1.35 + 0.742i)T + (0.535 - 0.844i)T^{2} \)
43 \( 1 + (0.951 - 0.309i)T^{2} \)
47 \( 1 + (0.481 + 0.876i)T^{2} \)
53 \( 1 + (-0.400 + 0.173i)T + (0.684 - 0.728i)T^{2} \)
59 \( 1 + (-0.0627 - 0.998i)T^{2} \)
61 \( 1 + (-1.74 - 0.961i)T + (0.535 + 0.844i)T^{2} \)
67 \( 1 + (0.904 - 0.425i)T^{2} \)
71 \( 1 + (0.876 - 0.481i)T^{2} \)
73 \( 1 + (-0.0540 + 1.72i)T + (-0.998 - 0.0627i)T^{2} \)
79 \( 1 + (0.992 + 0.125i)T^{2} \)
83 \( 1 + (0.125 + 0.992i)T^{2} \)
89 \( 1 + (1.23 + 1.31i)T + (-0.0627 + 0.998i)T^{2} \)
97 \( 1 + (0.436 - 1.95i)T + (-0.904 - 0.425i)T^{2} \)
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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

−8.812417774399156650407433007622, −7.69855555758460980729590990344, −6.93301279055503909094400413185, −6.46530090663555683690812610499, −5.60599373371858127891340378522, −4.78253412800832602722936549719, −4.13233933291251547941404418033, −2.79742748460050036153195778205, −2.27373952278604018903512389007, −0.923927910640083246920424549825, 1.33238240438493452323500456230, 2.40663841312632066953290398243, 2.95195749917509325243090907685, 4.31028928923584993768301284546, 5.15169915532339738420616685969, 5.54941526208267629478417721304, 6.52577756104116946032307709263, 7.19167664827508920665233782070, 8.135435194824706194904752879781, 8.469097518866360041246910705637

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