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Properties

Label 1-4011-4011.416-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.399 - 0.916i$
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  + (−0.202 − 0.979i)2-s + (−0.917 + 0.396i)4-s + (−0.298 − 0.954i)5-s + (0.574 + 0.818i)8-s + (−0.874 + 0.485i)10-s + (−0.350 − 0.936i)11-s + (0.277 + 0.960i)13-s + (0.685 − 0.728i)16-s + (0.840 − 0.542i)17-s + (−0.874 − 0.485i)19-s + (0.652 + 0.757i)20-s + (−0.846 + 0.533i)22-s + (0.992 − 0.120i)23-s + (−0.821 + 0.569i)25-s + (0.884 − 0.466i)26-s + ⋯
L(s)  = 1  + (−0.202 − 0.979i)2-s + (−0.917 + 0.396i)4-s + (−0.298 − 0.954i)5-s + (0.574 + 0.818i)8-s + (−0.874 + 0.485i)10-s + (−0.350 − 0.936i)11-s + (0.277 + 0.960i)13-s + (0.685 − 0.728i)16-s + (0.840 − 0.542i)17-s + (−0.874 − 0.485i)19-s + (0.652 + 0.757i)20-s + (−0.846 + 0.533i)22-s + (0.992 − 0.120i)23-s + (−0.821 + 0.569i)25-s + (0.884 − 0.466i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.399 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7037688586 - 1.073837790i\)
\(L(\frac12)\) \(\approx\) \(0.7037688586 - 1.073837790i\)
\(L(1)\) \(\approx\) \(0.7081015433 - 0.5192991901i\)
\(L(1)\) \(\approx\) \(0.7081015433 - 0.5192991901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.202 - 0.979i)T \)
5 \( 1 + (-0.298 - 0.954i)T \)
11 \( 1 + (-0.350 - 0.936i)T \)
13 \( 1 + (0.277 + 0.960i)T \)
17 \( 1 + (0.840 - 0.542i)T \)
19 \( 1 + (-0.874 - 0.485i)T \)
23 \( 1 + (0.992 - 0.120i)T \)
29 \( 1 + (0.518 + 0.854i)T \)
31 \( 1 + (0.962 - 0.272i)T \)
37 \( 1 + (-0.962 - 0.272i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (0.991 - 0.131i)T \)
47 \( 1 + (-0.0605 + 0.998i)T \)
53 \( 1 + (0.782 - 0.622i)T \)
59 \( 1 + (-0.556 - 0.831i)T \)
61 \( 1 + (0.256 - 0.966i)T \)
67 \( 1 + (0.159 + 0.987i)T \)
71 \( 1 + (0.213 + 0.976i)T \)
73 \( 1 + (0.968 + 0.250i)T \)
79 \( 1 + (-0.480 - 0.876i)T \)
83 \( 1 + (0.601 - 0.799i)T \)
89 \( 1 + (0.565 + 0.824i)T \)
97 \( 1 + (-0.371 + 0.928i)T \)
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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

−18.46590724390743305077282912495, −18.034778601374519082154823793897, −17.21517419427767097630881200797, −16.86689814906854203054099028275, −15.58481502488996263375776518083, −15.39470458239013863964289146465, −14.87801383594900249362953332047, −14.091476814529172505530748300946, −13.44662336299158063962177849852, −12.57436311622774907420532050558, −12.025860155442871307942065039752, −10.68378694149639535232778689538, −10.4001176493012542366526213587, −9.8113992146452031188632454413, −8.7126357431119375571960817306, −8.08519346907126690268647998887, −7.474551043041190357887390543515, −6.855658671453549766282922037611, −6.10074146840655227819216703975, −5.45236116648332639193964598246, −4.54954881798547587168153420277, −3.77638433322738174617150739551, −2.979131267166126355710969619765, −1.92402404012881248731881498502, −0.72168452420707260845772421820, 0.64167966541600597628579490860, 1.23806597190677727317996540103, 2.26379186499161758439495515128, 3.13999833014527443795697005520, 3.83506775083215864402239395686, 4.74659124668188166880982037832, 5.12106576920600427786420970009, 6.18260184048698386406820825, 7.21897463513511797736784869974, 8.16307299527337184723611176511, 8.64917397157428044944343661219, 9.19327467935386152852860400322, 9.94197582837865610060062340869, 10.88436665307120067570128537015, 11.33918088818748551401492877506, 12.07178515743434123355268700477, 12.6802984818809649233822872730, 13.33896444261594602273952223975, 13.933039466652128494203687184145, 14.64691581557191816871615094339, 15.84227413189946387779212726817, 16.29798776281146752092792865187, 17.03959166005546083972906937068, 17.51376833570710053379358561625, 18.589801079076172823069342559138

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