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Properties

Label 1-4000-4000.139-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.452 - 0.891i$
Analytic cond. $429.859$
Root an. cond. $429.859$
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  + (−0.790 + 0.612i)3-s + (0.587 − 0.809i)7-s + (0.248 − 0.968i)9-s + (0.995 − 0.0941i)11-s + (0.860 − 0.509i)13-s + (−0.187 + 0.982i)17-s + (−0.612 + 0.790i)19-s + (0.0314 + 0.999i)21-s + (0.844 − 0.535i)23-s + (0.397 + 0.917i)27-s + (0.338 + 0.940i)29-s + (0.187 − 0.982i)31-s + (−0.728 + 0.684i)33-s + (−0.917 − 0.397i)37-s + (−0.368 + 0.929i)39-s + ⋯
L(s)  = 1  + (−0.790 + 0.612i)3-s + (0.587 − 0.809i)7-s + (0.248 − 0.968i)9-s + (0.995 − 0.0941i)11-s + (0.860 − 0.509i)13-s + (−0.187 + 0.982i)17-s + (−0.612 + 0.790i)19-s + (0.0314 + 0.999i)21-s + (0.844 − 0.535i)23-s + (0.397 + 0.917i)27-s + (0.338 + 0.940i)29-s + (0.187 − 0.982i)31-s + (−0.728 + 0.684i)33-s + (−0.917 − 0.397i)37-s + (−0.368 + 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.452 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5357173327 - 0.8726720186i\)
\(L(\frac12)\) \(\approx\) \(0.5357173327 - 0.8726720186i\)
\(L(1)\) \(\approx\) \(0.9077899300 + 0.003109856286i\)
\(L(1)\) \(\approx\) \(0.9077899300 + 0.003109856286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.790 + 0.612i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.995 - 0.0941i)T \)
13 \( 1 + (0.860 - 0.509i)T \)
17 \( 1 + (-0.187 + 0.982i)T \)
19 \( 1 + (-0.612 + 0.790i)T \)
23 \( 1 + (0.844 - 0.535i)T \)
29 \( 1 + (0.338 + 0.940i)T \)
31 \( 1 + (0.187 - 0.982i)T \)
37 \( 1 + (-0.917 - 0.397i)T \)
41 \( 1 + (-0.844 - 0.535i)T \)
43 \( 1 + (-0.453 + 0.891i)T \)
47 \( 1 + (-0.876 + 0.481i)T \)
53 \( 1 + (-0.0314 - 0.999i)T \)
59 \( 1 + (-0.661 + 0.750i)T \)
61 \( 1 + (0.975 + 0.218i)T \)
67 \( 1 + (-0.940 - 0.338i)T \)
71 \( 1 + (-0.481 - 0.876i)T \)
73 \( 1 + (0.998 + 0.0627i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (0.612 - 0.790i)T \)
89 \( 1 + (0.998 + 0.0627i)T \)
97 \( 1 + (0.425 - 0.904i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.57385248899656105699419484228, −17.7551348282394431048441934358, −17.334358769116801273139061546368, −16.66194976412118139291049412220, −15.765771629249752572199965580769, −15.321019556403319801915932755375, −14.31513014931243006175795423040, −13.67947671648148232498122546540, −13.08899697988204471079899293195, −12.10089097392307645154420595073, −11.68246741548762818564944781053, −11.262340657729671162513022840428, −10.44666313582328681654739646619, −9.386774556679197256124332534500, −8.752984498726526274103137056062, −8.149371679513308050470239428, −6.98369027375187987155237512905, −6.705870204292938320351705197111, −5.89730996778312274627170747132, −5.0155561374931140301833069503, −4.604071097595079823028730449922, −3.44037599612060252802773216836, −2.40524857133465027678436068198, −1.620942727776049159941817227905, −0.97653620536731015034057878324, 0.18947285602771236474081233379, 1.15783393463390589061585629918, 1.71306936643025938981775188495, 3.34913360232807092415942549193, 3.812187562934946592477398022939, 4.52412048114437237490114958270, 5.236980658656639153479122506913, 6.26570556248802711112766300908, 6.50792850302198084988571444709, 7.55558654003000156974890822278, 8.50270707236993987471698489792, 8.9628789571901235925695525228, 10.13008770636998830766014551959, 10.48696423056562111744266785274, 11.1415816451077687833468535787, 11.710215839993239714902956003787, 12.60156354099679283029161470907, 13.18172610875313674111842092225, 14.18144267445397319247018255130, 14.8012914184088538798413553968, 15.2707517918436765089990575679, 16.37785547325245269784146748302, 16.654372625078245381739640835932, 17.41032435109542505910403463747, 17.796529654584515441356721561796

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