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Properties

Label 1-4000-4000.2053-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.969 - 0.244i$
Analytic cond. $429.859$
Root an. cond. $429.859$
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  + (0.917 + 0.397i)3-s + (0.309 + 0.951i)7-s + (0.684 + 0.728i)9-s + (−0.960 − 0.278i)11-s + (0.999 + 0.0314i)13-s + (−0.844 + 0.535i)17-s + (0.917 − 0.397i)19-s + (−0.0941 + 0.995i)21-s + (−0.992 − 0.125i)23-s + (0.338 + 0.940i)27-s + (0.860 − 0.509i)29-s + (0.535 + 0.844i)31-s + (−0.770 − 0.637i)33-s + (−0.940 − 0.338i)37-s + (0.904 + 0.425i)39-s + ⋯
L(s)  = 1  + (0.917 + 0.397i)3-s + (0.309 + 0.951i)7-s + (0.684 + 0.728i)9-s + (−0.960 − 0.278i)11-s + (0.999 + 0.0314i)13-s + (−0.844 + 0.535i)17-s + (0.917 − 0.397i)19-s + (−0.0941 + 0.995i)21-s + (−0.992 − 0.125i)23-s + (0.338 + 0.940i)27-s + (0.860 − 0.509i)29-s + (0.535 + 0.844i)31-s + (−0.770 − 0.637i)33-s + (−0.940 − 0.338i)37-s + (0.904 + 0.425i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.969 - 0.244i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.969 - 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1824158086 + 1.471851464i\)
\(L(\frac12)\) \(\approx\) \(-0.1824158086 + 1.471851464i\)
\(L(1)\) \(\approx\) \(1.245397877 + 0.4708105037i\)
\(L(1)\) \(\approx\) \(1.245397877 + 0.4708105037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.917 + 0.397i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.960 - 0.278i)T \)
13 \( 1 + (0.999 + 0.0314i)T \)
17 \( 1 + (-0.844 + 0.535i)T \)
19 \( 1 + (0.917 - 0.397i)T \)
23 \( 1 + (-0.992 - 0.125i)T \)
29 \( 1 + (0.860 - 0.509i)T \)
31 \( 1 + (0.535 + 0.844i)T \)
37 \( 1 + (-0.940 - 0.338i)T \)
41 \( 1 + (0.125 + 0.992i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (-0.998 + 0.0627i)T \)
53 \( 1 + (-0.995 - 0.0941i)T \)
59 \( 1 + (0.827 - 0.562i)T \)
61 \( 1 + (0.790 - 0.612i)T \)
67 \( 1 + (0.860 + 0.509i)T \)
71 \( 1 + (-0.998 + 0.0627i)T \)
73 \( 1 + (-0.187 - 0.982i)T \)
79 \( 1 + (-0.929 + 0.368i)T \)
83 \( 1 + (-0.397 - 0.917i)T \)
89 \( 1 + (-0.982 + 0.187i)T \)
97 \( 1 + (0.248 - 0.968i)T \)
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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

−17.889798434970034693048566944814, −17.65699129069645933010199469514, −16.403610125498706936826952648447, −15.77525412301806374317883678823, −15.356208599300152191146950245804, −14.23646280247296323691545152314, −13.84272971025066034037349415087, −13.38909471052089826509823187522, −12.655605811865410289164468504341, −11.7904537921901344439239322816, −11.030089487677081449170910627219, −10.12816607666338698644509971159, −9.79748112307729842265828587059, −8.555689044482109132709070998010, −8.31981123763316525228849923982, −7.37009039402746678720887204879, −7.01014162841756089883200559205, −6.03874468038305483613113265842, −5.052063057316664235155605129823, −4.19493070417192477540704225674, −3.56737926763627070964177600658, −2.73849373164841263103138565122, −1.87764071766010552567077048397, −1.11532715408034984649024235824, −0.173703295228030453694381972900, 1.297656610978074028336573401030, 2.10320035163035051776054251898, 2.86021415394737993935400685942, 3.448956568086122572958308451874, 4.49529994462285179195317038329, 5.06077925505266762370080313781, 5.962647775604299911648514115820, 6.71059979082843229016078085508, 7.872885792833821955222059234618, 8.32267609301676842300074984906, 8.76079779080386663586429847248, 9.663264705118399059048362802100, 10.274666211157796777642612396941, 11.10680216974733321165872755631, 11.69040642757566278263530697445, 12.774911098948559782167353533767, 13.22686841318933510763547399329, 14.01679630447419482250733256010, 14.53036556116914635365474917052, 15.49969166965968639649012914145, 15.86936797797781548707490542375, 16.14043833498351676253920139960, 17.6306635642675012363331207364, 18.018542891095063886499937983008, 18.7092843935304327407149997036

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