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Properties

Label 1-4011-4011.2477-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.890 - 0.454i$
Analytic cond. $431.041$
Root an. cond. $431.041$
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  + (−0.945 − 0.324i)2-s + (0.789 + 0.614i)4-s + (0.789 + 0.614i)5-s + (−0.546 − 0.837i)8-s + (−0.546 − 0.837i)10-s + (−0.879 − 0.475i)11-s + (0.401 − 0.915i)13-s + (0.245 + 0.969i)16-s + (−0.986 − 0.164i)17-s + (0.546 − 0.837i)19-s + (0.245 + 0.969i)20-s + (0.677 + 0.735i)22-s + (−0.546 + 0.837i)23-s + (0.245 + 0.969i)25-s + (−0.677 + 0.735i)26-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)2-s + (0.789 + 0.614i)4-s + (0.789 + 0.614i)5-s + (−0.546 − 0.837i)8-s + (−0.546 − 0.837i)10-s + (−0.879 − 0.475i)11-s + (0.401 − 0.915i)13-s + (0.245 + 0.969i)16-s + (−0.986 − 0.164i)17-s + (0.546 − 0.837i)19-s + (0.245 + 0.969i)20-s + (0.677 + 0.735i)22-s + (−0.546 + 0.837i)23-s + (0.245 + 0.969i)25-s + (−0.677 + 0.735i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.890 - 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132280237 - 0.2724860703i\)
\(L(\frac12)\) \(\approx\) \(1.132280237 - 0.2724860703i\)
\(L(1)\) \(\approx\) \(0.7238979177 - 0.06079050598i\)
\(L(1)\) \(\approx\) \(0.7238979177 - 0.06079050598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.945 - 0.324i)T \)
5 \( 1 + (0.789 + 0.614i)T \)
11 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (0.401 - 0.915i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
19 \( 1 + (0.546 - 0.837i)T \)
23 \( 1 + (-0.546 + 0.837i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (0.677 + 0.735i)T \)
41 \( 1 + (-0.945 + 0.324i)T \)
43 \( 1 + (-0.0825 - 0.996i)T \)
47 \( 1 + (0.879 + 0.475i)T \)
53 \( 1 + (-0.879 - 0.475i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.986 + 0.164i)T \)
67 \( 1 + (-0.986 + 0.164i)T \)
71 \( 1 + (0.945 - 0.324i)T \)
73 \( 1 + (-0.879 + 0.475i)T \)
79 \( 1 + (0.245 + 0.969i)T \)
83 \( 1 + (-0.546 - 0.837i)T \)
89 \( 1 + (0.0825 - 0.996i)T \)
97 \( 1 + (0.677 + 0.735i)T \)
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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

−18.21752011397066582713757973341, −17.86405744416507351525042897060, −16.88420894528706396731266662156, −16.4857359520646461485021348695, −15.89260823935089209590204600829, −15.13647975277464266623160779191, −14.27002029282225159856973840786, −13.7474935115863250944538550244, −12.78672043574737607632156898752, −12.23068817491010585812602837869, −11.24217631466889769712163618638, −10.59640602119143576738429513739, −9.94270994562351953260621302204, −9.25811626224728449817209440716, −8.74049490315582232528345072643, −7.98569871382227098590136119075, −7.24548639894232426861902435145, −6.35904797608039675609881700017, −5.88357195392650125022067657424, −4.99958469416498730845838229647, −4.29188888547591469086126856511, −2.94857391347972066959667002971, −1.95629390756876454475911089974, −1.679851186702560203122388688359, −0.482469428838782351346751259, 0.40357876444372029939378943959, 1.38587988463781879765915639590, 2.29214377798621229589249983335, 2.918191879715789881483768343082, 3.50804934686400328931857889269, 4.83468676449353043997425546784, 5.78243700927294537713033231912, 6.31387535705207351097629413032, 7.24116989829048603102631454285, 7.77248785550067313820611616428, 8.65412369521530898400255955100, 9.25824813063771317905405266612, 10.09003470222069194376302515582, 10.53185970019511854151135053634, 11.1904841264464753481283519529, 11.77706960785077825158281353326, 12.91189241767881931874134266169, 13.37901735580610879797219574895, 13.967528761555396311827082399453, 15.32988618430808719341240309381, 15.45586313055934884555723877218, 16.260405569955750791661000805087, 17.2956874022551161141763989777, 17.59170434214844689523275597662, 18.434289002440260027027733302152

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