Properties

Degree 1
Conductor 4001
Sign $-0.151 + 0.988i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.309 − 0.951i)2-s + (0.587 + 0.809i)3-s + (−0.809 − 0.587i)4-s + 5-s + (0.951 − 0.309i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)9-s + (0.309 − 0.951i)10-s i·11-s i·12-s + (−0.809 − 0.587i)13-s + 14-s + (0.587 + 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + ⋯
L(s,χ)  = 1  + (0.309 − 0.951i)2-s + (0.587 + 0.809i)3-s + (−0.809 − 0.587i)4-s + 5-s + (0.951 − 0.309i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)9-s + (0.309 − 0.951i)10-s i·11-s i·12-s + (−0.809 − 0.587i)13-s + 14-s + (0.587 + 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.151 + 0.988i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.151 + 0.988i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4001\)
\( \varepsilon \)  =  $-0.151 + 0.988i$
motivic weight  =  \(0\)
character  :  $\chi_{4001} (1070, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4001,\ (0:\ ),\ -0.151 + 0.988i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7506656369 + 0.8749054495i$
$L(\frac12,\chi)$  $\approx$  $0.7506656369 + 0.8749054495i$
$L(\chi,1)$  $\approx$  1.251642196 - 0.1284186164i
$L(1,\chi)$  $\approx$  1.251642196 - 0.1284186164i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.15781983632743695726118542816, −17.52623715286893029257304145889, −16.87629497896679118926435008706, −16.67345740382398064980342398708, −15.15293477807115185410497419256, −14.78620499140633925212865433705, −14.26256818168375436117751385946, −13.644160295242621918035772276982, −12.93162104077735356943666786917, −12.61792640562209975282252648007, −11.70912353934475356690769014706, −10.410072307672853585513632000858, −9.858331308362653994456730679838, −9.080190916462602716051187233, −8.365802929559121945255494701448, −7.69422834585084957713863205943, −6.85262415119254739999947205567, −6.64539347257537096026005990439, −5.779870584548652265291782568805, −4.73983036234104936303895115338, −4.20678480553714232057755935893, −3.29002248228578328763314448864, −1.984175559585756636626998387845, −1.82352891987811590753863587198, −0.2296688346626364113614866625, 1.33465401094908097337175564604, 2.39343406513159888280012677749, 2.60374734065427060041935415705, 3.377365718392113923907559594902, 4.49216444503718678908483536117, 5.05582393703796425441027341679, 5.6531318615623849913563254221, 6.3463664781811852254192544088, 7.84874048698846562981058132124, 8.56781579943385277029826267110, 9.272347693996309617502162414749, 9.5581903968162716660863752390, 10.40206689702418156734914206896, 11.06848163763095930434517358858, 11.619251522317547381197936273548, 12.59730652910266597246343269381, 13.32869825966323446285896001741, 13.80849970839191706643032758944, 14.48995239999143463415791265489, 15.096185766479866500737125073716, 15.697918269395931134292361741061, 16.69271644639913261967569016064, 17.486589699671594168950712622771, 18.10146870834757135942178690730, 18.77437559160348847431507015901

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