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Properties

Label 1-3997-3997.804-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.735 - 0.677i$
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  + (−0.191 − 0.981i)2-s + (0.360 + 0.932i)3-s + (−0.926 + 0.376i)4-s + (0.319 + 0.947i)5-s + (0.846 − 0.533i)6-s + (0.546 + 0.837i)8-s + (−0.739 + 0.673i)9-s + (0.868 − 0.495i)10-s + (0.00551 + 0.999i)11-s + (−0.685 − 0.728i)12-s + (0.381 − 0.924i)13-s + (−0.768 + 0.639i)15-s + (0.716 − 0.697i)16-s + (0.989 − 0.142i)17-s + (0.802 + 0.596i)18-s + (−0.775 + 0.631i)19-s + ⋯
L(s)  = 1  + (−0.191 − 0.981i)2-s + (0.360 + 0.932i)3-s + (−0.926 + 0.376i)4-s + (0.319 + 0.947i)5-s + (0.846 − 0.533i)6-s + (0.546 + 0.837i)8-s + (−0.739 + 0.673i)9-s + (0.868 − 0.495i)10-s + (0.00551 + 0.999i)11-s + (−0.685 − 0.728i)12-s + (0.381 − 0.924i)13-s + (−0.768 + 0.639i)15-s + (0.716 − 0.697i)16-s + (0.989 − 0.142i)17-s + (0.802 + 0.596i)18-s + (−0.775 + 0.631i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (804, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04859481142 + 0.1245734853i\)
\(L(\frac12)\) \(\approx\) \(-0.04859481142 + 0.1245734853i\)
\(L(1)\) \(\approx\) \(0.9395595116 + 0.1331966095i\)
\(L(1)\) \(\approx\) \(0.9395595116 + 0.1331966095i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.191 - 0.981i)T \)
3 \( 1 + (0.360 + 0.932i)T \)
5 \( 1 + (0.319 + 0.947i)T \)
11 \( 1 + (0.00551 + 0.999i)T \)
13 \( 1 + (0.381 - 0.924i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (0.965 + 0.261i)T \)
29 \( 1 + (0.0275 + 0.999i)T \)
31 \( 1 + (-0.245 - 0.969i)T \)
37 \( 1 + (-0.997 + 0.0770i)T \)
41 \( 1 + (0.298 - 0.954i)T \)
43 \( 1 + (0.411 - 0.911i)T \)
47 \( 1 + (0.962 + 0.272i)T \)
53 \( 1 + (0.731 - 0.681i)T \)
59 \( 1 + (0.986 - 0.164i)T \)
61 \( 1 + (0.857 + 0.514i)T \)
67 \( 1 + (0.224 + 0.974i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.565 - 0.824i)T \)
79 \( 1 + (-0.899 + 0.436i)T \)
83 \( 1 + (0.0385 + 0.999i)T \)
89 \( 1 + (-0.884 - 0.466i)T \)
97 \( 1 + (-0.528 + 0.849i)T \)
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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

−17.73943399566457758071219150575, −17.15943290014102590262791752740, −16.62482946120207271901677007065, −16.05012692956507931360152330199, −15.150813113974582206456131784292, −14.30860547025088728913260476887, −13.89543413557611189620999147337, −13.2044184330257230181678634750, −12.75447312951172374350794960007, −11.90705298506183830522866551699, −11.04636913890628428275599361279, −9.990174459119656377051147693919, −9.08942703202679148163541804357, −8.71393872328975389354425916300, −8.23627792209626822960146731069, −7.36074824783822862914123350553, −6.63626945227774692459165703605, −5.9842733260057591512079810163, −5.41037456988760338870573189457, −4.47195733556055659179380333718, −3.6546268362761968807363404343, −2.57343636770913364734766670002, −1.38555513754098247454043801946, −0.961783686855422315678534162887, −0.02183228946934187405353722094, 1.31921888201143402356778928250, 2.2947341574882545652638492733, 2.82326252905725106700437186214, 3.693967109743260327988884488295, 4.04911023348919294900361937159, 5.35964752903793495443719367683, 5.53878240424977008753872196685, 7.03502660318362346339726219460, 7.69388650264738987361947719882, 8.56383368810930532040255502735, 9.22392874373653439237410554516, 10.06406585518394425868550723786, 10.349639060280802028591140084225, 10.87507272894275624664692557011, 11.70375854077651416678405862741, 12.530039964005599942700389773768, 13.22918905108637805277227510963, 14.02789541834392755590330630091, 14.697829853129134629134063039730, 15.05000182109954562618654901104, 15.96594871105538190268558474164, 16.96721380966103914388352889309, 17.431839782579183588643752210289, 18.12156976999309400815830722178, 19.0063542772512501380490927707

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