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Properties

Label 1-10e3-1000.797-r1-0-0
Degree $1$
Conductor $1000$
Sign $0.910 + 0.414i$
Analytic cond. $107.464$
Root an. cond. $107.464$
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  + (−0.248 + 0.968i)3-s + (−0.951 + 0.309i)7-s + (−0.876 − 0.481i)9-s + (0.187 + 0.982i)11-s + (−0.481 + 0.876i)13-s + (0.368 − 0.929i)17-s + (0.968 − 0.248i)19-s + (−0.0627 − 0.998i)21-s + (0.904 + 0.425i)23-s + (0.684 − 0.728i)27-s + (−0.637 − 0.770i)29-s + (−0.929 − 0.368i)31-s + (−0.998 − 0.0627i)33-s + (−0.684 − 0.728i)37-s + (−0.728 − 0.684i)39-s + ⋯
L(s)  = 1  + (−0.248 + 0.968i)3-s + (−0.951 + 0.309i)7-s + (−0.876 − 0.481i)9-s + (0.187 + 0.982i)11-s + (−0.481 + 0.876i)13-s + (0.368 − 0.929i)17-s + (0.968 − 0.248i)19-s + (−0.0627 − 0.998i)21-s + (0.904 + 0.425i)23-s + (0.684 − 0.728i)27-s + (−0.637 − 0.770i)29-s + (−0.929 − 0.368i)31-s + (−0.998 − 0.0627i)33-s + (−0.684 − 0.728i)37-s + (−0.728 − 0.684i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.910 + 0.414i$
Analytic conductor: \(107.464\)
Root analytic conductor: \(107.464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1000,\ (1:\ ),\ 0.910 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137066773 + 0.2466735940i\)
\(L(\frac12)\) \(\approx\) \(1.137066773 + 0.2466735940i\)
\(L(1)\) \(\approx\) \(0.7809972602 + 0.2712856159i\)
\(L(1)\) \(\approx\) \(0.7809972602 + 0.2712856159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.248 + 0.968i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.187 + 0.982i)T \)
13 \( 1 + (-0.481 + 0.876i)T \)
17 \( 1 + (0.368 - 0.929i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (0.904 + 0.425i)T \)
29 \( 1 + (-0.637 - 0.770i)T \)
31 \( 1 + (-0.929 - 0.368i)T \)
37 \( 1 + (-0.684 - 0.728i)T \)
41 \( 1 + (-0.425 - 0.904i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.844 + 0.535i)T \)
53 \( 1 + (-0.998 + 0.0627i)T \)
59 \( 1 + (-0.992 + 0.125i)T \)
61 \( 1 + (0.425 - 0.904i)T \)
67 \( 1 + (0.770 + 0.637i)T \)
71 \( 1 + (0.535 - 0.844i)T \)
73 \( 1 + (-0.125 + 0.992i)T \)
79 \( 1 + (-0.968 - 0.248i)T \)
83 \( 1 + (0.248 + 0.968i)T \)
89 \( 1 + (0.992 + 0.125i)T \)
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

−21.65172115805069096216349332126, −20.244989126859866693086018230785, −19.861916725744600944134526293795, −18.90102032990638879602235804296, −18.53855826892172301502294662467, −17.41495127771391965293425294481, −16.77178701503953737158757914387, −16.16915421226447622681154760256, −14.96666816636965605726659540576, −14.153061998785557742450219842162, −13.24009526837467246813146197274, −12.78511542056238074123606392066, −11.98768690267952600431440480412, −10.98875186360585124771789406286, −10.29025204566030234158698786600, −9.1842967271415286426713924376, −8.26930780996024024675891367905, −7.45114786698823585310510656724, −6.641261035839665379779227281318, −5.85657222851997316944399877208, −5.10473115592656044976516083299, −3.405335053345114145260649296156, −3.03155433427365740720879151286, −1.53344449275183605680089975033, −0.6426491117995179736473290104, 0.401257927848888204103906277638, 2.08574847321624613933675285378, 3.144854854361957037467326468103, 3.95254817810997916855663557603, 4.98200013070892093359964765798, 5.61841330049436560906143078261, 6.81219085909123614428933381944, 7.432011967237604884187001766674, 9.06223794273984635451968308429, 9.40860156314158229243605174843, 9.9991732256663991130054244113, 11.12816762890709089767445406250, 11.88921712632869871286331492111, 12.55560946377622372423926697742, 13.72966258210500349324917778055, 14.494163217218434772260155418000, 15.455992679649924356734918012926, 15.86205358332499317155172245618, 16.82470607643640880315899882186, 17.3078627996687137391872417863, 18.45218439610656728087263306491, 19.190954568710191357568376010052, 20.186643204726035551154220192767, 20.65472320830324757610571360381, 21.665343810197683822105200051269

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