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Properties

Label 1-3997-3997.538-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.892 - 0.450i$
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  + (−0.298 − 0.954i)2-s + (−0.984 + 0.175i)3-s + (−0.821 + 0.569i)4-s + (−0.565 − 0.824i)5-s + (0.461 + 0.887i)6-s + (0.789 + 0.614i)8-s + (0.938 − 0.345i)9-s + (−0.618 + 0.785i)10-s + (−0.989 + 0.142i)11-s + (0.709 − 0.705i)12-s + (−0.731 − 0.681i)13-s + (0.701 + 0.712i)15-s + (0.350 − 0.936i)16-s + (0.834 − 0.551i)17-s + (−0.609 − 0.792i)18-s + (−0.471 − 0.882i)19-s + ⋯
L(s)  = 1  + (−0.298 − 0.954i)2-s + (−0.984 + 0.175i)3-s + (−0.821 + 0.569i)4-s + (−0.565 − 0.824i)5-s + (0.461 + 0.887i)6-s + (0.789 + 0.614i)8-s + (0.938 − 0.345i)9-s + (−0.618 + 0.785i)10-s + (−0.989 + 0.142i)11-s + (0.709 − 0.705i)12-s + (−0.731 − 0.681i)13-s + (0.701 + 0.712i)15-s + (0.350 − 0.936i)16-s + (0.834 − 0.551i)17-s + (−0.609 − 0.792i)18-s + (−0.471 − 0.882i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.892 - 0.450i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.892 - 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1444103229 - 0.6063295392i\)
\(L(\frac12)\) \(\approx\) \(0.1444103229 - 0.6063295392i\)
\(L(1)\) \(\approx\) \(0.4196397331 - 0.2887839663i\)
\(L(1)\) \(\approx\) \(0.4196397331 - 0.2887839663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.298 - 0.954i)T \)
3 \( 1 + (-0.984 + 0.175i)T \)
5 \( 1 + (-0.565 - 0.824i)T \)
11 \( 1 + (-0.989 + 0.142i)T \)
13 \( 1 + (-0.731 - 0.681i)T \)
17 \( 1 + (0.834 - 0.551i)T \)
19 \( 1 + (-0.471 - 0.882i)T \)
23 \( 1 + (0.828 + 0.560i)T \)
29 \( 1 + (-0.754 + 0.656i)T \)
31 \( 1 + (0.986 - 0.164i)T \)
37 \( 1 + (-0.421 - 0.906i)T \)
41 \( 1 + (-0.0275 + 0.999i)T \)
43 \( 1 + (-0.0385 + 0.999i)T \)
47 \( 1 + (-0.635 - 0.771i)T \)
53 \( 1 + (0.802 - 0.596i)T \)
59 \( 1 + (0.401 - 0.915i)T \)
61 \( 1 + (-0.0935 - 0.995i)T \)
67 \( 1 + (-0.917 - 0.396i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.996 - 0.0880i)T \)
79 \( 1 + (0.685 + 0.728i)T \)
83 \( 1 + (0.537 - 0.843i)T \)
89 \( 1 + (-0.999 - 0.0440i)T \)
97 \( 1 + (-0.329 + 0.944i)T \)
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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

−18.5855779954746458159114366565, −17.88537464067913156439156834166, −17.05127746633826132503593985134, −16.712932735246658720842140585650, −15.96534218792988527976742947815, −15.23938667123631049489688332939, −14.81639003006738820273041453529, −13.95299942994742701069642634486, −13.218479606132130766964037166940, −12.349914077570814523288528940654, −11.81161386887618691151699287579, −10.760184392907183670708885630689, −10.35169553943056044887394977471, −9.82771127852325528930227776564, −8.605773608846737378365692825179, −7.89544820328134441518699670896, −7.28689476245369938817552343169, −6.76388434147034951843194717845, −5.95615193636210086504284525250, −5.4106184269681909456767586358, −4.504420547322137802366611462894, −3.912475146852767104320682825990, −2.6815156088975459020527556725, −1.579282508829988914966538606072, −0.49113924503339129547397894471, 0.31104552059418741766915029801, 0.79391974003033755180598731412, 1.792894163726630917811082337907, 2.89423801546375840563979201170, 3.6030843817969467639375470305, 4.71667324390246427331243789053, 4.97474767811758683226770662696, 5.572888702599103134561369136837, 6.990035480832749975029058890825, 7.6604573998608414592607531888, 8.26057287870239095514275848057, 9.3193384169242461623816289747, 9.75128986258430316979240221360, 10.56913694629388657682151396397, 11.17622736413323184194522490534, 11.78477958321862582239409057998, 12.45676428873421651879947500423, 12.96863385110442461078628182719, 13.408996679196093459782581407136, 14.77660856015287370602676112080, 15.43141158656557330867674736968, 16.26672589999760565568969024186, 16.740477991858800951640359669876, 17.49405684688107985768358654599, 17.93676624938929552136936493961

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