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Properties

Label 1-4000-4000.77-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.650 + 0.759i$
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 + (−0.809 − 0.587i)7-s + (0.248 − 0.968i)9-s + (−0.0941 − 0.995i)11-s + (0.860 − 0.509i)13-s + (0.982 + 0.187i)17-s + (0.790 + 0.612i)19-s + (−0.999 + 0.0314i)21-s + (0.535 + 0.844i)23-s + (−0.397 − 0.917i)27-s + (−0.940 + 0.338i)29-s + (−0.187 + 0.982i)31-s + (−0.684 − 0.728i)33-s + (−0.917 − 0.397i)37-s + (0.368 − 0.929i)39-s + ⋯
L(s)  = 1  + (0.790 − 0.612i)3-s + (−0.809 − 0.587i)7-s + (0.248 − 0.968i)9-s + (−0.0941 − 0.995i)11-s + (0.860 − 0.509i)13-s + (0.982 + 0.187i)17-s + (0.790 + 0.612i)19-s + (−0.999 + 0.0314i)21-s + (0.535 + 0.844i)23-s + (−0.397 − 0.917i)27-s + (−0.940 + 0.338i)29-s + (−0.187 + 0.982i)31-s + (−0.684 − 0.728i)33-s + (−0.917 − 0.397i)37-s + (0.368 − 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2615816441 - 0.5685885772i\)
\(L(\frac12)\) \(\approx\) \(-0.2615816441 - 0.5685885772i\)
\(L(1)\) \(\approx\) \(1.107948913 - 0.4557384982i\)
\(L(1)\) \(\approx\) \(1.107948913 - 0.4557384982i\)

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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.0941 - 0.995i)T \)
13 \( 1 + (0.860 - 0.509i)T \)
17 \( 1 + (0.982 + 0.187i)T \)
19 \( 1 + (0.790 + 0.612i)T \)
23 \( 1 + (0.535 + 0.844i)T \)
29 \( 1 + (-0.940 + 0.338i)T \)
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 \( 1 + (-0.481 - 0.876i)T \)
53 \( 1 + (0.0314 + 0.999i)T \)
59 \( 1 + (-0.750 - 0.661i)T \)
61 \( 1 + (0.218 - 0.975i)T \)
67 \( 1 + (-0.940 - 0.338i)T \)
71 \( 1 + (-0.481 - 0.876i)T \)
73 \( 1 + (0.0627 - 0.998i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (-0.612 + 0.790i)T \)
89 \( 1 + (0.998 + 0.0627i)T \)
97 \( 1 + (-0.904 - 0.425i)T \)
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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

−18.85561946755625186366738813538, −18.31228949244213906315516101933, −17.28974913617213846792498609967, −16.43954701457356876931645387085, −16.01900350504421561888630337990, −15.29826192410170284792653361816, −14.81256788414658733994811558975, −14.02903174725267753223989040427, −13.28885025440758317395063130448, −12.75756623900370453416414342510, −11.88496443764141464644071795947, −11.15085389599721859546300004135, −10.16767764567075375851560172704, −9.74426739183630567628009471959, −9.04670867064388831415257697727, −8.56329428091522085631793193197, −7.49524441830646244651943681583, −7.03096248844305384827208935651, −5.94274664635235851827609938989, −5.26001286965041623241983931978, −4.376548615783362272190184513505, −3.66947649757954746418598710187, −2.91772215846326494765033564234, −2.26055119985514274660480927298, −1.299679374208702786689660540380, 0.07869484679547288488635535240, 1.11636935031114407476767819625, 1.52983974615447440507546138704, 3.09949805232409397669796741129, 3.25663344665911783394794038904, 3.85512972442083019375920384118, 5.286419548380478619991962540722, 5.97944485921624660648568956918, 6.639109133798952704636304385, 7.59672510651926075211504774671, 7.88693036369785717973208784388, 8.87566256165260449356877733328, 9.39207016754545701301581914878, 10.2373800015589526538674835238, 10.91436263183546907824244976472, 11.80919591253573792094434679074, 12.64305358268357501098755101025, 13.14035719257451495867852494477, 13.8018968551648346549047918817, 14.23108652695270223684594804481, 15.10708898298555534382735523700, 15.932055420507118961198874355553, 16.40422548347757796741743097914, 17.191732465153131669397604290476, 18.168400506365648283613017193113

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