↑

Properties

Label 1-4031-4031.28-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.696 - 0.717i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.715 − 0.699i)2-s + (0.949 − 0.313i)3-s + (0.0227 − 0.999i)4-s + (−0.829 + 0.557i)5-s + (0.460 − 0.887i)6-s + (0.538 − 0.842i)7-s + (−0.682 − 0.730i)8-s + (0.803 − 0.595i)9-s + (−0.203 + 0.979i)10-s + (0.648 + 0.761i)11-s + (−0.291 − 0.956i)12-s + (0.746 + 0.665i)13-s + (−0.203 − 0.979i)14-s + (−0.613 + 0.789i)15-s + (−0.998 − 0.0455i)16-s + (0.419 − 0.907i)17-s + ⋯
L(s)  = 1  + (0.715 − 0.699i)2-s + (0.949 − 0.313i)3-s + (0.0227 − 0.999i)4-s + (−0.829 + 0.557i)5-s + (0.460 − 0.887i)6-s + (0.538 − 0.842i)7-s + (−0.682 − 0.730i)8-s + (0.803 − 0.595i)9-s + (−0.203 + 0.979i)10-s + (0.648 + 0.761i)11-s + (−0.291 − 0.956i)12-s + (0.746 + 0.665i)13-s + (−0.203 − 0.979i)14-s + (−0.613 + 0.789i)15-s + (−0.998 − 0.0455i)16-s + (0.419 − 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.364769598 - 3.225171920i\)
\(L(\frac12)\) \(\approx\) \(1.364769598 - 3.225171920i\)
\(L(1)\) \(\approx\) \(1.595137355 - 1.200602758i\)
\(L(1)\) \(\approx\) \(1.595137355 - 1.200602758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.715 - 0.699i)T \)
3 \( 1 + (0.949 - 0.313i)T \)
5 \( 1 + (-0.829 + 0.557i)T \)
7 \( 1 + (0.538 - 0.842i)T \)
11 \( 1 + (0.648 + 0.761i)T \)
13 \( 1 + (0.746 + 0.665i)T \)
17 \( 1 + (0.419 - 0.907i)T \)
19 \( 1 + (-0.995 + 0.0909i)T \)
23 \( 1 + (0.460 + 0.887i)T \)
31 \( 1 + (-0.803 - 0.595i)T \)
37 \( 1 + (-0.613 - 0.789i)T \)
41 \( 1 + (-0.934 - 0.356i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.898 + 0.439i)T \)
53 \( 1 + (-0.247 - 0.968i)T \)
59 \( 1 + (0.854 - 0.519i)T \)
61 \( 1 + (-0.934 + 0.356i)T \)
67 \( 1 + (0.983 + 0.181i)T \)
71 \( 1 + (0.898 + 0.439i)T \)
73 \( 1 + (0.974 - 0.225i)T \)
79 \( 1 + (0.775 + 0.631i)T \)
83 \( 1 + (0.983 - 0.181i)T \)
89 \( 1 + (-0.377 - 0.926i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.795845168218767880962160465426, −18.08360036211330053540028877879, −16.94678813398976093255758410594, −16.56762450645056992774685574642, −15.75942325620785025995603152575, −15.19071408352824999533456479574, −14.826397151778128846764644556405, −14.163708601601719144495513450016, −13.29289489099675811837144445026, −12.67728792853055226484881935743, −12.19310833530498752417567576033, −11.20802764408834326796284256165, −10.67209140841801805860171122938, −9.26387773540449944159092504573, −8.58705281340568250635435967577, −8.35965380692733516806075403839, −7.84345844976053358713237202511, −6.73142625845615329038226805137, −6.02454488682404020582659441414, −5.062220709477556598295348198170, −4.57521759317448033083350541986, −3.53247396016304696754783523720, −3.4158409463051199199494904880, −2.260047674036881433304418512754, −1.247577540807498071312412374493, 0.67212910576674129915342967724, 1.69796324687098544926660543722, 2.16741676893332041622097818976, 3.38966503097498682163546912649, 3.75350319915381650126956755974, 4.30211043372802523214547616853, 5.15241411368236955658079343039, 6.522827965426325674467169987384, 6.967004646295337259868064702630, 7.571374372138275722375211061319, 8.50911765476032993330762293026, 9.30811849840351915186287824728, 9.97531714319840879769340338925, 10.85226839022242364885232307978, 11.38412237863231553827580385302, 12.05293259903791730301668808460, 12.78834172408579602109710288009, 13.535190710621726402039146913922, 14.23843128664086196528128737752, 14.48966088507961436981853690203, 15.253723816822888552565743408865, 15.80248207137921451780521839928, 16.812876445035062000105189800428, 17.86914827830053522969561109287, 18.478213235455580697531834838299

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