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Properties

Label 1-3997-3997.832-r1-0-0
Degree $1$
Conductor $3997$
Sign $0.951 + 0.307i$
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  + (0.635 + 0.771i)2-s + (0.609 − 0.792i)3-s + (−0.191 + 0.981i)4-s + (0.949 + 0.314i)5-s + (0.999 − 0.0330i)6-s + (−0.879 + 0.475i)8-s + (−0.256 − 0.966i)9-s + (0.360 + 0.932i)10-s + (0.988 − 0.153i)11-s + (0.660 + 0.750i)12-s + (0.0385 + 0.999i)13-s + (0.828 − 0.560i)15-s + (−0.926 − 0.376i)16-s + (0.644 − 0.764i)17-s + (0.583 − 0.812i)18-s + (−0.959 + 0.282i)19-s + ⋯
L(s)  = 1  + (0.635 + 0.771i)2-s + (0.609 − 0.792i)3-s + (−0.191 + 0.981i)4-s + (0.949 + 0.314i)5-s + (0.999 − 0.0330i)6-s + (−0.879 + 0.475i)8-s + (−0.256 − 0.966i)9-s + (0.360 + 0.932i)10-s + (0.988 − 0.153i)11-s + (0.660 + 0.750i)12-s + (0.0385 + 0.999i)13-s + (0.828 − 0.560i)15-s + (−0.926 − 0.376i)16-s + (0.644 − 0.764i)17-s + (0.583 − 0.812i)18-s + (−0.959 + 0.282i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $0.951 + 0.307i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ 0.951 + 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.767626951 + 0.9088529074i\)
\(L(\frac12)\) \(\approx\) \(5.767626951 + 0.9088529074i\)
\(L(1)\) \(\approx\) \(2.150436040 + 0.5154544010i\)
\(L(1)\) \(\approx\) \(2.150436040 + 0.5154544010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.635 + 0.771i)T \)
3 \( 1 + (0.609 - 0.792i)T \)
5 \( 1 + (0.949 + 0.314i)T \)
11 \( 1 + (0.988 - 0.153i)T \)
13 \( 1 + (0.0385 + 0.999i)T \)
17 \( 1 + (0.644 - 0.764i)T \)
19 \( 1 + (-0.959 + 0.282i)T \)
23 \( 1 + (0.431 + 0.901i)T \)
29 \( 1 + (0.716 - 0.697i)T \)
31 \( 1 + (-0.789 + 0.614i)T \)
37 \( 1 + (-0.556 - 0.831i)T \)
41 \( 1 + (0.592 - 0.805i)T \)
43 \( 1 + (0.775 - 0.631i)T \)
47 \( 1 + (-0.137 - 0.990i)T \)
53 \( 1 + (-0.537 - 0.843i)T \)
59 \( 1 + (0.0825 - 0.996i)T \)
61 \( 1 + (0.834 - 0.551i)T \)
67 \( 1 + (0.999 - 0.0440i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.441 - 0.897i)T \)
79 \( 1 + (0.996 - 0.0880i)T \)
83 \( 1 + (-0.471 + 0.882i)T \)
89 \( 1 + (-0.528 - 0.849i)T \)
97 \( 1 + (0.992 - 0.120i)T \)
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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

−18.45733611341016771204625624806, −17.52329123490056524697913529709, −16.95835270354164858314192114344, −16.173268084427829258202524813, −15.23228804864462222545681508273, −14.63218331640662781382931273956, −14.33203935922979867842605242612, −13.49404466615310200566865619412, −12.72515809578311508110120024407, −12.46050421732486474347746362685, −11.14354299226137349111825338775, −10.68021380709186429541224986047, −10.01878165022831449944753252009, −9.46463344400729216853309289518, −8.77917826272181101236975325842, −8.1566606281906109739343328309, −6.76290640084493575267959881166, −6.01079114463356545056101557237, −5.36916210389324516575881933892, −4.55413953619367436301677968975, −4.036116517256629499849307517930, −3.055269635397548563318053979057, −2.540332885563601626615133762911, −1.5993778242662977221365055460, −0.88249750134682553985193336910, 0.65479810950345432380650521475, 1.856203604738598159879125754261, 2.30660354503369525034645927019, 3.45031015698479055526758107686, 3.840320050672864779565439617489, 5.06204584030125684232730278665, 5.77101016616937341503424516649, 6.66196986130680547688130579261, 6.789381885008431476501192502115, 7.65314311601476581043953926206, 8.56823971931366056264474110879, 9.19483478493484848190764998998, 9.61851814934172231184666775667, 10.96368741336706208861715895733, 11.77884093486103541942510247944, 12.40115399235438526415454973236, 13.05696015830818996085138539873, 13.91970431596064195357097422806, 14.139647663632130914553260812158, 14.58164822994271597663311197349, 15.4714187087455198839836270, 16.33631458029591949698824946565, 17.16454688028043815780403006249, 17.443820816873118434506256414962, 18.27396348419691227928174194210

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