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Properties

Label 1-3997-3997.615-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: $\chi_{3997} (615, \cdot )$
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

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

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