Properties

Label 1-4011-4011.5-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.998 - 0.0521i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (−0.0275 − 0.999i)2-s + (−0.998 + 0.0550i)4-s + (0.451 − 0.892i)5-s + (0.0825 + 0.996i)8-s + (−0.904 − 0.426i)10-s + (0.298 + 0.954i)11-s + (0.986 − 0.164i)13-s + (0.993 − 0.110i)16-s + (0.716 + 0.697i)17-s + (−0.904 + 0.426i)19-s + (−0.401 + 0.915i)20-s + (0.945 − 0.324i)22-s + (−0.904 + 0.426i)23-s + (−0.592 − 0.805i)25-s + (−0.191 − 0.981i)26-s + ⋯
L(s)  = 1  + (−0.0275 − 0.999i)2-s + (−0.998 + 0.0550i)4-s + (0.451 − 0.892i)5-s + (0.0825 + 0.996i)8-s + (−0.904 − 0.426i)10-s + (0.298 + 0.954i)11-s + (0.986 − 0.164i)13-s + (0.993 − 0.110i)16-s + (0.716 + 0.697i)17-s + (−0.904 + 0.426i)19-s + (−0.401 + 0.915i)20-s + (0.945 − 0.324i)22-s + (−0.904 + 0.426i)23-s + (−0.592 − 0.805i)25-s + (−0.191 − 0.981i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.998 - 0.0521i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.998 - 0.0521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.405908284 - 0.03671422600i\)
\(L(\frac12)\) \(\approx\) \(1.405908284 - 0.03671422600i\)
\(L(1)\) \(\approx\) \(0.9433572432 - 0.4096941188i\)
\(L(1)\) \(\approx\) \(0.9433572432 - 0.4096941188i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.0275 - 0.999i)T \)
5 \( 1 + (0.451 - 0.892i)T \)
11 \( 1 + (0.298 + 0.954i)T \)
13 \( 1 + (0.986 - 0.164i)T \)
17 \( 1 + (0.716 + 0.697i)T \)
19 \( 1 + (-0.904 + 0.426i)T \)
23 \( 1 + (-0.904 + 0.426i)T \)
29 \( 1 + (0.401 + 0.915i)T \)
31 \( 1 + (0.754 - 0.656i)T \)
37 \( 1 + (-0.754 - 0.656i)T \)
41 \( 1 + (-0.879 + 0.475i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (0.975 + 0.218i)T \)
53 \( 1 + (0.298 + 0.954i)T \)
59 \( 1 + (-0.754 + 0.656i)T \)
61 \( 1 + (-0.716 + 0.697i)T \)
67 \( 1 + (-0.962 - 0.272i)T \)
71 \( 1 + (0.879 - 0.475i)T \)
73 \( 1 + (-0.975 + 0.218i)T \)
79 \( 1 + (0.993 - 0.110i)T \)
83 \( 1 + (-0.0825 - 0.996i)T \)
89 \( 1 + (-0.926 - 0.376i)T \)
97 \( 1 + (-0.945 + 0.324i)T \)
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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

−18.40719672769885921268054555248, −17.77056748260596906829147013520, −17.05824977486494996989537209103, −16.48984324499713974589420962481, −15.61147846328883286825792211118, −15.30142638438980850745819046149, −14.16451697177367891616755763180, −13.94172246707268587631864113853, −13.49165394976726711716395309931, −12.40180590620096911525148957185, −11.60523399023559764986476321630, −10.6880422347776779667559442015, −10.18184930104045139676821272107, −9.34700315131779928756767952245, −8.55768423567576788256151337465, −8.09134947733088543133133148669, −7.05568159795611129455936144345, −6.49703143418025442479502421154, −5.98636809276923831504502729946, −5.28813357558473946109100449600, −4.23024043205485248504455042671, −3.537109862274722383625393091242, −2.75553994157812303372628279153, −1.56719402802887591412748770659, −0.43607181535624922663297334874, 1.10179364932008206429784311022, 1.58316008411536654400364725485, 2.346017137520447524042486842517, 3.460741934269958844988014018558, 4.18533973871794637033610718530, 4.698092169990887697880906715210, 5.75638503958035976695381593093, 6.14375022367978387352789280652, 7.550033241678532020078392850391, 8.27117132405826248580170919186, 8.89022655833977075077385076613, 9.51237034827412103153749776732, 10.34420211567888929582786966600, 10.64704751081509307168846397502, 11.88689451018523361945484937344, 12.2510899421498094505752732651, 12.84198268249654440821664138552, 13.54212447840822953182097780125, 14.14214766532024246960665018879, 14.94242829653701482177511873442, 15.784157475642303790126655864968, 16.74232956671257757956754602499, 17.186394155909939244022488107301, 17.901667329951062936125304740267, 18.43075754166803042440958331269

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