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Properties

Label 1-3997-3997.16-r0-0-0
Degree $1$
Conductor $3997$
Sign $-0.895 - 0.444i$
Analytic cond. $18.5619$
Root an. cond. $18.5619$
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  + (0.945 + 0.324i)2-s + (0.546 + 0.837i)3-s + (0.789 + 0.614i)4-s + (−0.677 − 0.735i)5-s + (0.245 + 0.969i)6-s + (0.546 + 0.837i)8-s + (−0.401 + 0.915i)9-s + (−0.401 − 0.915i)10-s + (−0.401 + 0.915i)11-s + (−0.0825 + 0.996i)12-s + (0.0275 + 0.999i)13-s + (0.245 − 0.969i)15-s + (0.245 + 0.969i)16-s + (0.716 − 0.697i)17-s + (−0.677 + 0.735i)18-s + (−0.998 + 0.0550i)19-s + ⋯
L(s)  = 1  + (0.945 + 0.324i)2-s + (0.546 + 0.837i)3-s + (0.789 + 0.614i)4-s + (−0.677 − 0.735i)5-s + (0.245 + 0.969i)6-s + (0.546 + 0.837i)8-s + (−0.401 + 0.915i)9-s + (−0.401 − 0.915i)10-s + (−0.401 + 0.915i)11-s + (−0.0825 + 0.996i)12-s + (0.0275 + 0.999i)13-s + (0.245 − 0.969i)15-s + (0.245 + 0.969i)16-s + (0.716 − 0.697i)17-s + (−0.677 + 0.735i)18-s + (−0.998 + 0.0550i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(18.5619\)
Root analytic conductor: \(18.5619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (0:\ ),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4065229475 + 1.731661356i\)
\(L(\frac12)\) \(\approx\) \(-0.4065229475 + 1.731661356i\)
\(L(1)\) \(\approx\) \(1.293999693 + 0.9651040607i\)
\(L(1)\) \(\approx\) \(1.293999693 + 0.9651040607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.945 + 0.324i)T \)
3 \( 1 + (0.546 + 0.837i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
11 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (0.0275 + 0.999i)T \)
17 \( 1 + (0.716 - 0.697i)T \)
19 \( 1 + (-0.998 + 0.0550i)T \)
23 \( 1 + (-0.998 - 0.0550i)T \)
29 \( 1 + (0.851 - 0.523i)T \)
31 \( 1 + (-0.962 - 0.272i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (0.975 - 0.218i)T \)
43 \( 1 + (-0.592 + 0.805i)T \)
47 \( 1 + (-0.962 - 0.272i)T \)
53 \( 1 + (0.945 - 0.324i)T \)
59 \( 1 + (0.350 - 0.936i)T \)
61 \( 1 + (-0.754 + 0.656i)T \)
67 \( 1 + (-0.754 - 0.656i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.879 + 0.475i)T \)
79 \( 1 + (0.137 + 0.990i)T \)
83 \( 1 + (0.716 - 0.697i)T \)
89 \( 1 + (0.245 + 0.969i)T \)
97 \( 1 + (0.137 - 0.990i)T \)
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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

−18.28920759105675420408106770885, −17.69089625540577750574580413666, −16.47092598825558162637392017821, −15.85079561032701386598372060563, −15.01457896628395549707820155262, −14.66120881261941444572468112734, −13.98453712575803723462919802682, −13.319094224795197071487298135138, −12.588162891009901716232581300, −12.163725343986956227875637650031, −11.36505886131838183557818329509, −10.53227080414722764313369634713, −10.26493673724605189363242070067, −8.81941844236455307497785318083, −8.11341143089275531044397775415, −7.54151154410771428130745335944, −6.7876172786625022914259984474, −6.01429235502575809759592429875, −5.55772053427862446437987023834, −4.26084300978811336428344144438, −3.49283421521979916189454394311, −3.08343995494226363936658561950, −2.320830729673859129606348920151, −1.40534189118530210258977712862, −0.28510614526119197371884333970, 1.70616141269600799630830497930, 2.373882069808533714983363646057, 3.37533660983307073211103114065, 4.058703928957832078292300743358, 4.6117023044088220896639691191, 5.067694569941641959416273020026, 6.01534308287814246721690661568, 7.02181191434875840618517527980, 7.74934959838420777856595753752, 8.28610324617768337082675584182, 9.07641297022268212250917887267, 9.88226213434608535545568185168, 10.65004159752115826956784347511, 11.59356555813468342526504448096, 12.01983849783106348796549867283, 12.80869041080688871800223229717, 13.501851842727828541435444696481, 14.247423196104447409065131410273, 14.8304756032829442615079509638, 15.413455537074937905240243593998, 16.12820159087999091377594325658, 16.452328100693783438076736168814, 17.105981342369256377763927704244, 18.10983335328053707309367607500, 19.29156431143954117668907382928

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