↑

Properties

Label 1-4003-4003.1004-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.622 + 0.782i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (0.961 + 0.275i)2-s + (−0.999 − 0.0172i)3-s + (0.847 + 0.530i)4-s + (−0.896 + 0.443i)5-s + (−0.956 − 0.292i)6-s + (0.862 − 0.506i)7-s + (0.668 + 0.743i)8-s + (0.999 + 0.0345i)9-s + (−0.983 + 0.179i)10-s + (0.997 − 0.0658i)11-s + (−0.838 − 0.544i)12-s + (0.999 + 0.0376i)13-s + (0.968 − 0.248i)14-s + (0.903 − 0.428i)15-s + (0.437 + 0.899i)16-s + (0.294 − 0.955i)17-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (−0.999 − 0.0172i)3-s + (0.847 + 0.530i)4-s + (−0.896 + 0.443i)5-s + (−0.956 − 0.292i)6-s + (0.862 − 0.506i)7-s + (0.668 + 0.743i)8-s + (0.999 + 0.0345i)9-s + (−0.983 + 0.179i)10-s + (0.997 − 0.0658i)11-s + (−0.838 − 0.544i)12-s + (0.999 + 0.0376i)13-s + (0.968 − 0.248i)14-s + (0.903 − 0.428i)15-s + (0.437 + 0.899i)16-s + (0.294 − 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.622 + 0.782i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.725217593 + 1.314697613i\)
\(L(\frac12)\) \(\approx\) \(2.725217593 + 1.314697613i\)
\(L(1)\) \(\approx\) \(1.614741324 + 0.4470490466i\)
\(L(1)\) \(\approx\) \(1.614741324 + 0.4470490466i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
3 \( 1 + (-0.999 - 0.0172i)T \)
5 \( 1 + (-0.896 + 0.443i)T \)
7 \( 1 + (0.862 - 0.506i)T \)
11 \( 1 + (0.997 - 0.0658i)T \)
13 \( 1 + (0.999 + 0.0376i)T \)
17 \( 1 + (0.294 - 0.955i)T \)
19 \( 1 + (0.744 + 0.668i)T \)
23 \( 1 + (-0.0932 + 0.995i)T \)
29 \( 1 + (0.380 + 0.924i)T \)
31 \( 1 + (0.981 + 0.193i)T \)
37 \( 1 + (-0.416 + 0.909i)T \)
41 \( 1 + (0.943 + 0.332i)T \)
43 \( 1 + (-0.940 - 0.340i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.597 - 0.802i)T \)
59 \( 1 + (0.733 + 0.679i)T \)
61 \( 1 + (-0.136 - 0.990i)T \)
67 \( 1 + (0.181 - 0.983i)T \)
71 \( 1 + (-0.967 + 0.253i)T \)
73 \( 1 + (0.451 + 0.892i)T \)
79 \( 1 + (0.000785 - 0.999i)T \)
83 \( 1 + (-0.919 + 0.392i)T \)
89 \( 1 + (-0.761 - 0.647i)T \)
97 \( 1 + (0.999 - 0.0219i)T \)
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

−18.50945083225275750956507814413, −17.54437702250064543802537152907, −16.98955771532840693322721875855, −16.06489595036336129327139543403, −15.73515329073526039688660157758, −15.02145269770010273975357755879, −14.3573811996918155054156853775, −13.47320029648215302463229976667, −12.684983307027580665126995025160, −12.03783065207598651013672956529, −11.73613988842089965040191724251, −11.05162200896365439411692695946, −10.57209121498362077218603197801, −9.476117323358089050312433659853, −8.551984761294602218154043017918, −7.77582419468606607747517614652, −6.928876739864794718873153537984, −6.14043512835802005857173426469, −5.63731395999715212616827518679, −4.70664466683391316736013854843, −4.26184772948982773079016952218, −3.676674534161634124591473226932, −2.47476485685259446605322946350, −1.32749179019314152480366755534, −0.94681955648768876219417922395, 1.04772971126466173175940639975, 1.60515256668718268065648367751, 3.135713274572975912510880799490, 3.71885098441500276025329871454, 4.38643015375689159031873678234, 5.05832057978082338116111105714, 5.792485536176506693151416593308, 6.695481895293906594593575303234, 7.088043381653481082656069146, 7.82429424433190825600930748875, 8.52450477017939134251106589831, 9.863656571736702323185208557206, 10.7019434051609035740949123113, 11.302715796254028881316949598362, 11.82843797348666239017119619997, 12.03252510347272834612606420511, 13.16890546142663560790072213696, 13.98649732340191755091970272259, 14.31895819297978621984688980642, 15.31158659538220535472730975831, 15.821633761835777657846703443737, 16.42779326250380041515935670508, 17.01990304692131904323697798090, 17.828214864663657649173535525199, 18.40317681076342269771352627974

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