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Properties

Label 1-3997-3997.13-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.989 - 0.146i$
Analytic cond. $429.537$
Root an. cond. $429.537$
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  + (−0.926 − 0.376i)2-s + (0.739 + 0.673i)3-s + (0.716 + 0.697i)4-s + (0.795 + 0.605i)5-s + (−0.431 − 0.901i)6-s + (−0.401 − 0.915i)8-s + (0.0935 + 0.995i)9-s + (−0.509 − 0.860i)10-s + (−0.999 − 0.0110i)11-s + (0.0605 + 0.998i)12-s + (0.709 − 0.705i)13-s + (0.180 + 0.983i)15-s + (0.0275 + 0.999i)16-s + (−0.959 − 0.282i)17-s + (0.287 − 0.957i)18-s + (−0.202 − 0.979i)19-s + ⋯
L(s)  = 1  + (−0.926 − 0.376i)2-s + (0.739 + 0.673i)3-s + (0.716 + 0.697i)4-s + (0.795 + 0.605i)5-s + (−0.431 − 0.901i)6-s + (−0.401 − 0.915i)8-s + (0.0935 + 0.995i)9-s + (−0.509 − 0.860i)10-s + (−0.999 − 0.0110i)11-s + (0.0605 + 0.998i)12-s + (0.709 − 0.705i)13-s + (0.180 + 0.983i)15-s + (0.0275 + 0.999i)16-s + (−0.959 − 0.282i)17-s + (0.287 − 0.957i)18-s + (−0.202 − 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.989 - 0.146i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04829084445 + 0.6571473464i\)
\(L(\frac12)\) \(\approx\) \(-0.04829084445 + 0.6571473464i\)
\(L(1)\) \(\approx\) \(0.8821408931 + 0.2127163660i\)
\(L(1)\) \(\approx\) \(0.8821408931 + 0.2127163660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.926 - 0.376i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (0.795 + 0.605i)T \)
11 \( 1 + (-0.999 - 0.0110i)T \)
13 \( 1 + (0.709 - 0.705i)T \)
17 \( 1 + (-0.959 - 0.282i)T \)
19 \( 1 + (-0.202 - 0.979i)T \)
23 \( 1 + (0.863 - 0.504i)T \)
29 \( 1 + (-0.998 - 0.0550i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (0.988 + 0.153i)T \)
41 \( 1 + (0.821 - 0.569i)T \)
43 \( 1 + (-0.660 + 0.750i)T \)
47 \( 1 + (-0.851 + 0.523i)T \)
53 \( 1 + (0.0715 + 0.997i)T \)
59 \( 1 + (-0.945 - 0.324i)T \)
61 \( 1 + (-0.471 + 0.882i)T \)
67 \( 1 + (-0.899 - 0.436i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.360 + 0.932i)T \)
79 \( 1 + (0.618 + 0.785i)T \)
83 \( 1 + (0.997 + 0.0770i)T \)
89 \( 1 + (-0.565 + 0.824i)T \)
97 \( 1 + (0.441 - 0.897i)T \)
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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

−18.10054603682583908094682338555, −17.395358798471200564945822985634, −16.69062176382396981687484144778, −16.069165694823139068318032904614, −15.20583625075784475943348815944, −14.716754577804401587996070772108, −13.72813722991827522922954808588, −13.32993595870056623251689559265, −12.66530148604529963746767835652, −11.68825700966614401016642015155, −10.93207452157433861016008648663, −10.09296885032538051153769265664, −9.35981202490167828585753903155, −8.93799563947094107510314963411, −8.15152801768377058790418347285, −7.74152604990621098428041605497, −6.687324189619369139520526605940, −6.22838837665857268534634702145, −5.47916666735042877807198419427, −4.49397317866828037360807915228, −3.319620340223141885036416897845, −2.28551047911285771984105604143, −1.84965619314646395271944663144, −1.094707234583433934549230408500, −0.11397682772077103856941899197, 1.09615268554394613569427970442, 2.16373616710538424088523653183, 2.79060516948640706861640367367, 3.10589478835067671130703550318, 4.28434343730313418509466534527, 5.142396394881619366940302987, 6.14380306899490687522593052570, 6.91355364400731619117265784298, 7.73614891002493443600496766230, 8.34994355098780362064468593947, 9.18768074267884673403109408308, 9.51771193605711703829609715822, 10.46799213504774749221930720923, 10.86237537423627390284184212546, 11.22958869862078263043294088760, 12.70358309928942750768586301305, 13.269321180387750758142308860167, 13.70080663597756681643072154580, 14.904899495839269936435025408, 15.32385588457054545714770229471, 15.90114577180134518411008584262, 16.71094637697283848914309014990, 17.44172178832278438547391471703, 18.19733380611048481378370882073, 18.48423019961471086298104310525

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