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Properties

Label 1-4000-4000.1117-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.244 + 0.969i$
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  + (0.509 − 0.860i)3-s + (0.309 − 0.951i)7-s + (−0.481 − 0.876i)9-s + (0.562 + 0.827i)11-s + (−0.278 + 0.960i)13-s + (0.368 + 0.929i)17-s + (0.509 + 0.860i)19-s + (−0.661 − 0.750i)21-s + (−0.425 − 0.904i)23-s + (−0.999 − 0.0314i)27-s + (0.995 − 0.0941i)29-s + (−0.929 + 0.368i)31-s + (0.998 − 0.0627i)33-s + (−0.0314 − 0.999i)37-s + (0.684 + 0.728i)39-s + ⋯
L(s)  = 1  + (0.509 − 0.860i)3-s + (0.309 − 0.951i)7-s + (−0.481 − 0.876i)9-s + (0.562 + 0.827i)11-s + (−0.278 + 0.960i)13-s + (0.368 + 0.929i)17-s + (0.509 + 0.860i)19-s + (−0.661 − 0.750i)21-s + (−0.425 − 0.904i)23-s + (−0.999 − 0.0314i)27-s + (0.995 − 0.0941i)29-s + (−0.929 + 0.368i)31-s + (0.998 − 0.0627i)33-s + (−0.0314 − 0.999i)37-s + (0.684 + 0.728i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.244 + 0.969i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.244 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.091556579 + 0.8508250127i\)
\(L(\frac12)\) \(\approx\) \(1.091556579 + 0.8508250127i\)
\(L(1)\) \(\approx\) \(1.167229464 - 0.2659814410i\)
\(L(1)\) \(\approx\) \(1.167229464 - 0.2659814410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.509 - 0.860i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.562 + 0.827i)T \)
13 \( 1 + (-0.278 + 0.960i)T \)
17 \( 1 + (0.368 + 0.929i)T \)
19 \( 1 + (0.509 + 0.860i)T \)
23 \( 1 + (-0.425 - 0.904i)T \)
29 \( 1 + (0.995 - 0.0941i)T \)
31 \( 1 + (-0.929 + 0.368i)T \)
37 \( 1 + (-0.0314 - 0.999i)T \)
41 \( 1 + (-0.904 - 0.425i)T \)
43 \( 1 + (0.156 + 0.987i)T \)
47 \( 1 + (0.844 - 0.535i)T \)
53 \( 1 + (-0.750 + 0.661i)T \)
59 \( 1 + (-0.790 + 0.612i)T \)
61 \( 1 + (0.338 - 0.940i)T \)
67 \( 1 + (0.995 + 0.0941i)T \)
71 \( 1 + (0.844 - 0.535i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (0.968 - 0.248i)T \)
83 \( 1 + (-0.860 + 0.509i)T \)
89 \( 1 + (-0.125 - 0.992i)T \)
97 \( 1 + (-0.770 - 0.637i)T \)
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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

−18.186837651051114394031210343243, −17.455655520365316989924264874607, −16.72984642734010241458723169451, −15.869303660927496739173792954421, −15.542224879475993262057203332601, −14.88228686618862187633887464638, −14.05773026524906082253386307956, −13.69001127378551539539160665414, −12.69635042665910087301095885135, −11.73139302271497009658254853808, −11.39815682544829102330168121671, −10.52297784193228933847307880591, −9.6638968794465696831126326686, −9.24320623233948742250543930862, −8.45381642199999206506973902723, −7.95006778583938381793583648183, −7.02059790157233401146051898293, −5.89351477921852250657392909512, −5.30326105995724818530703494246, −4.793404274772418407125292986376, −3.66350868586387138330031514244, −3.03838898114285076788869677277, −2.48025750101461348356088706952, −1.32467756868312172344407413353, −0.18578990832425161726521184610, 1.00897406787953263458070276376, 1.65409596215118004799482216251, 2.26848494464351910842107543414, 3.50545780980100199102483158500, 4.01715606932674030067927322062, 4.82748442453748694350817379872, 6.00638332927925451961822290562, 6.66134008345142473457001219385, 7.27848000074693363056614990947, 7.86942940412532061913769701031, 8.60383615624549821746291346504, 9.4042616658991571637831590914, 10.11205268613311501852197328720, 10.86317002063840015715185881804, 11.84557390552564940549678355619, 12.33571837256427954675185793578, 12.89255548017745454261494474579, 13.938410120482556267065391069057, 14.29102452694625352167908014563, 14.66259234875059766760576697745, 15.701876258759851697319471257238, 16.72015152983785883117961039408, 17.06639142228086967757256008457, 17.84168455314471163877487111075, 18.46681341765553499398611260620

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