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Properties

Label 1-3997-3997.3022-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.181 - 0.983i$
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  + (−0.5 − 0.866i)2-s + (−0.913 + 0.406i)3-s + (−0.5 + 0.866i)4-s + (−0.669 + 0.743i)5-s + (0.809 + 0.587i)6-s + 8-s + (0.669 − 0.743i)9-s + (0.978 + 0.207i)10-s + (0.669 + 0.743i)11-s + (0.104 − 0.994i)12-s + (−0.309 − 0.951i)13-s + (0.309 − 0.951i)15-s + (−0.5 − 0.866i)16-s + (−0.669 − 0.743i)17-s + (−0.978 − 0.207i)18-s + (−0.669 + 0.743i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.913 + 0.406i)3-s + (−0.5 + 0.866i)4-s + (−0.669 + 0.743i)5-s + (0.809 + 0.587i)6-s + 8-s + (0.669 − 0.743i)9-s + (0.978 + 0.207i)10-s + (0.669 + 0.743i)11-s + (0.104 − 0.994i)12-s + (−0.309 − 0.951i)13-s + (0.309 − 0.951i)15-s + (−0.5 − 0.866i)16-s + (−0.669 − 0.743i)17-s + (−0.978 − 0.207i)18-s + (−0.669 + 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.181 - 0.983i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (3022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2784665633 - 0.3344590496i\)
\(L(\frac12)\) \(\approx\) \(0.2784665633 - 0.3344590496i\)
\(L(1)\) \(\approx\) \(0.5056040320 - 0.05506519915i\)
\(L(1)\) \(\approx\) \(0.5056040320 - 0.05506519915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−18.4412031574657832434787052629, −17.580423326031056278866700685, −17.02742056119603824204648965838, −16.5636214764881273424489783248, −16.09252914587397074255245228683, −15.28599991272758680668622325839, −14.63608440703859931300242041985, −13.53610740254961345678538827809, −13.27418739405146553527054990100, −12.099891662336185582269864223155, −11.78090508355214137998849445028, −10.84249720854711888842527956581, −10.3051014241156089911655900339, −9.24053714290944534834469059998, −8.437469533307572770501622534247, −8.31244320352099537086197441728, −6.89199381809122810393305318901, −6.79753724395950804904043642563, −6.01768175496661043939683369681, −5.018162749248820026478754001207, −4.559062642942259076737128952601, −3.883715983072877391990909145171, −2.204757618741742318120365320897, −1.27896441014154520941046414116, −0.5930427016289464393384042548, 0.17298171016709920983964695889, 0.998777130433048109876687357161, 2.099792663640407798329893264383, 2.97860188661503604911013394211, 3.82935087646586138042462353417, 4.3491779495437911261599076418, 5.1221024680169244617968286211, 6.24511251342499694871613120029, 6.96514554600754504782460818838, 7.65254958566260737494407525082, 8.391310413892465827717481833430, 9.52046554287057467942123341747, 9.90003192267228179368587510311, 10.60810310323334273318144079139, 11.238447584690650460048682471702, 11.787951973094777263643440160359, 12.3223973096584353876619229076, 12.99511758300459641797391122661, 13.995156927765665140918754957723, 14.89654866051104424433034757001, 15.50232659846657630434258393603, 16.147039042929679532509666635242, 17.09332153975564828053946904783, 17.4919027146716114017206600241, 18.14787707455804393440139360145

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