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Properties

Label 1-4011-4011.3695-r1-0-0
Degree $1$
Conductor $4011$
Sign $-0.647 + 0.762i$
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  + (0.0825 + 0.996i)2-s + (−0.986 + 0.164i)4-s + (−0.986 + 0.164i)5-s + (−0.245 − 0.969i)8-s + (−0.245 − 0.969i)10-s + (0.789 + 0.614i)11-s + (0.879 − 0.475i)13-s + (0.945 − 0.324i)16-s + (−0.677 + 0.735i)17-s + (0.245 − 0.969i)19-s + (0.945 − 0.324i)20-s + (−0.546 + 0.837i)22-s + (−0.245 + 0.969i)23-s + (0.945 − 0.324i)25-s + (0.546 + 0.837i)26-s + ⋯
L(s)  = 1  + (0.0825 + 0.996i)2-s + (−0.986 + 0.164i)4-s + (−0.986 + 0.164i)5-s + (−0.245 − 0.969i)8-s + (−0.245 − 0.969i)10-s + (0.789 + 0.614i)11-s + (0.879 − 0.475i)13-s + (0.945 − 0.324i)16-s + (−0.677 + 0.735i)17-s + (0.245 − 0.969i)19-s + (0.945 − 0.324i)20-s + (−0.546 + 0.837i)22-s + (−0.245 + 0.969i)23-s + (0.945 − 0.324i)25-s + (0.546 + 0.837i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.647 + 0.762i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (3695, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ -0.647 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7274376208 + 1.571593192i\)
\(L(\frac12)\) \(\approx\) \(0.7274376208 + 1.571593192i\)
\(L(1)\) \(\approx\) \(0.7637346981 + 0.5256417859i\)
\(L(1)\) \(\approx\) \(0.7637346981 + 0.5256417859i\)

Euler product

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

−18.42519162120001990260701168338, −17.55307529933857620188282591248, −16.56985780387359423416817186567, −16.199266745433557924440090184718, −15.28018465721646717834221147133, −14.43465128901838292649143356001, −13.90727367719698970155503616588, −13.21618200393361861636316380606, −12.353201740700631829284945061680, −11.75561836688611557494427509529, −11.3747367482485716110947834364, −10.66016200691308857929427834244, −9.83550137260295677475836698180, −8.962753102458922563299316057403, −8.50332265111505211828732348145, −7.8690515374253904474798464374, −6.71306350280501645931995499358, −6.0424988538780576098496665922, −4.97378326098254568516196618547, −4.224951083194942638836850471542, −3.76656591601782608016790092029, −3.00602676278367151693721161415, −2.03950665792014473812182086858, −1.035802900750980163777867435921, −0.44653643069153265317528077316, 0.64399450342165483960326498407, 1.53087490102323515781474895963, 3.066396380996864550170194480019, 3.65054117479907015670806196352, 4.433633669708593003574344453837, 4.98732788102093761046402041337, 6.04869139803531389243948263442, 6.7536543619520157750319618064, 7.18098569651582737446503580568, 8.1425065893848035378970839622, 8.58093265574051844572665194858, 9.272572234424450969896801821891, 10.23017160009249476308421346959, 10.96086332820872607441275697318, 11.88272433230817412034396706443, 12.37571007207844045553586773964, 13.32841600978659741133326328952, 13.792452243569945233180271839591, 14.734886773695606650383630871921, 15.32460843039328854648237175522, 15.65861128377445172974304321088, 16.35239630157455722620171772577, 17.219782007944092277624807424441, 17.81246279839883249098339260269, 18.25535478951965886166640750752

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