Properties

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

Related objects

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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  :  $\chi_{4001} (1305, \cdot )$
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

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

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