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Properties

Label 1-4001-4001.2931-r0-0-0
Degree $1$
Conductor $4001$
Sign $0.999 + 0.00979i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
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  + (0.309 − 0.951i)2-s + (−0.587 − 0.809i)3-s + (−0.809 − 0.587i)4-s + 5-s + (−0.951 + 0.309i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)9-s + (0.309 − 0.951i)10-s + i·11-s + i·12-s + (−0.809 − 0.587i)13-s + 14-s + (−0.587 − 0.809i)15-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.587 − 0.809i)3-s + (−0.809 − 0.587i)4-s + 5-s + (−0.951 + 0.309i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)9-s + (0.309 − 0.951i)10-s + i·11-s + i·12-s + (−0.809 − 0.587i)13-s + 14-s + (−0.587 − 0.809i)15-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $0.999 + 0.00979i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (2931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ 0.999 + 0.00979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208918779 + 0.005918310248i\)
\(L(\frac12)\) \(\approx\) \(1.208918779 + 0.005918310248i\)
\(L(1)\) \(\approx\) \(0.8871554178 - 0.4738062740i\)
\(L(1)\) \(\approx\) \(0.8871554178 - 0.4738062740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 - iT \)
59 \( 1 - T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−18.2272670814182425725292702762, −17.3903239975585559082744382315, −16.93731891012950632484979589761, −16.61575617359804502885869095456, −15.9955747330392276707867062463, −14.975068498558394765666642581547, −14.34690191509483437583124521316, −14.03194993620027093935064276971, −13.20747768834440528747752508932, −12.46365127821650414025460213630, −11.554311633286165852112275150342, −10.79999622183024436451299159459, −10.11901150775229347067974427646, −9.30745012372295580177114289270, −9.00361315253226842299019020039, −7.81439288766287633909135745520, −7.13953218223057597366653554698, −6.33745245161109627502231289005, −5.743219426943952704647415915225, −5.07378316577172599674451384127, −4.503004338076983140057187918623, −3.63216064927239017187538272418, −2.94314935205615472172455520299, −1.47690471629536535126100892772, −0.35643325327187537402344898499, 1.02921770268570024189660066967, 1.917689321112609372823494849709, 2.296349821010894795806010808053, 2.98034347278327670263900495591, 4.47106954078948958456368242438, 5.03885224823198281641410723248, 5.656339978522338379858232454719, 6.25604771865605324912595461492, 7.13514513019248027959610720819, 8.16012858236317334617175451769, 8.89842453938029514760261901477, 9.61284678825894375158514014885, 10.52892394843179935018910666517, 10.74230175869177936210757726149, 11.91702240919889493024117608609, 12.37636153631504200392949529308, 12.80017089919149271697550126435, 13.36063258511291655122629368974, 14.4411588685890616609597228643, 14.69477734075465377657919682525, 15.55820786000914737282743893832, 16.92737797205266000064324986703, 17.43996657267018719623565071132, 17.82123992569224397010500580220, 18.45540443857269409714824366576

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