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Properties

Label 1-3997-3997.174-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: $\chi_{3997} (174, \cdot )$
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

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

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