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Properties

Label 1-4001-4001.1305-r0-0-0
Degree $1$
Conductor $4001$
Sign $-0.151 - 0.988i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
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  + (0.309 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $-0.151 - 0.988i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (1305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ -0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7506656369 - 0.8749054495i\)
\(L(\frac12)\) \(\approx\) \(0.7506656369 - 0.8749054495i\)
\(L(1)\) \(\approx\) \(1.251642196 + 0.1284186164i\)
\(L(1)\) \(\approx\) \(1.251642196 + 0.1284186164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 - iT \)
59 \( 1 - T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−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