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Properties

Label 1-4011-4011.53-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.549 + 0.835i$
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  + (0.868 − 0.495i)2-s + (0.509 − 0.860i)4-s + (0.298 + 0.954i)5-s + (0.0165 − 0.999i)8-s + (0.731 + 0.681i)10-s + (0.350 + 0.936i)11-s + (−0.340 + 0.940i)13-s + (−0.480 − 0.876i)16-s + (0.360 − 0.932i)17-s + (−0.731 + 0.681i)19-s + (0.973 + 0.229i)20-s + (0.768 + 0.639i)22-s + (0.421 + 0.906i)23-s + (−0.821 + 0.569i)25-s + (0.170 + 0.985i)26-s + ⋯
L(s)  = 1  + (0.868 − 0.495i)2-s + (0.509 − 0.860i)4-s + (0.298 + 0.954i)5-s + (0.0165 − 0.999i)8-s + (0.731 + 0.681i)10-s + (0.350 + 0.936i)11-s + (−0.340 + 0.940i)13-s + (−0.480 − 0.876i)16-s + (0.360 − 0.932i)17-s + (−0.731 + 0.681i)19-s + (0.973 + 0.229i)20-s + (0.768 + 0.639i)22-s + (0.421 + 0.906i)23-s + (−0.821 + 0.569i)25-s + (0.170 + 0.985i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.495197978 + 1.345817873i\)
\(L(\frac12)\) \(\approx\) \(2.495197978 + 1.345817873i\)
\(L(1)\) \(\approx\) \(1.765771358 + 0.02348116165i\)
\(L(1)\) \(\approx\) \(1.765771358 + 0.02348116165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.868 - 0.495i)T \)
5 \( 1 + (0.298 + 0.954i)T \)
11 \( 1 + (0.350 + 0.936i)T \)
13 \( 1 + (-0.340 + 0.940i)T \)
17 \( 1 + (0.360 - 0.932i)T \)
19 \( 1 + (-0.731 + 0.681i)T \)
23 \( 1 + (0.421 + 0.906i)T \)
29 \( 1 + (0.922 + 0.386i)T \)
31 \( 1 + (0.962 - 0.272i)T \)
37 \( 1 + (0.962 + 0.272i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (-0.724 + 0.689i)T \)
47 \( 1 + (-0.537 - 0.843i)T \)
53 \( 1 + (-0.834 - 0.551i)T \)
59 \( 1 + (-0.618 + 0.785i)T \)
61 \( 1 + (-0.840 - 0.542i)T \)
67 \( 1 + (-0.709 - 0.705i)T \)
71 \( 1 + (0.746 + 0.665i)T \)
73 \( 1 + (-0.930 + 0.366i)T \)
79 \( 1 + (-0.126 + 0.991i)T \)
83 \( 1 + (-0.574 - 0.818i)T \)
89 \( 1 + (-0.942 - 0.335i)T \)
97 \( 1 + (-0.846 + 0.533i)T \)
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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

−18.13850111942609898752478750461, −17.23137790421689595358805200778, −17.06316905111142219373119666647, −16.3256858907370736816204949571, −15.56024394150851350566758636695, −15.04890342649559333362417105180, −14.16294187119323289082654046190, −13.63659517474240908014120019269, −12.83806905400823872926347704440, −12.520023267003669696151127962968, −11.75968052644229948815673408636, −10.8665463712582653044597315776, −10.213326264669746212935782250401, −9.084763601327870718404346343874, −8.36254601650076882999514985009, −8.09870241965736427730496728919, −6.97138117642041926306830096355, −6.07339073988214155296901757659, −5.81740339934252807151162436876, −4.71196672407233579931325136935, −4.444245306893093181154191002345, −3.31675904110304528524308480678, −2.68979261885082715583336357565, −1.632388491136627105035925723855, −0.53874097670988085083093481706, 1.32503323374760259731334537945, 1.97218600371487254610616536409, 2.80501697748382581736151872932, 3.40174014436831194825080603926, 4.44152354141398647662493401778, 4.84588876382406966066869730745, 5.94010761501326495663174060512, 6.59014014990398205188372708559, 7.051234991450993645501341118070, 7.903726137998619174812246538709, 9.23559718240609319432652863264, 9.89674536296010316013176604065, 10.1928903991419662143466346585, 11.35350955886481450351851397039, 11.60533891653612259122692066739, 12.36909706509316976996944048881, 13.19200685841034773049503645192, 13.90050327066304255149633613848, 14.42503398033166681302562481872, 14.96730870833829524018680562659, 15.55946906179462868002356554146, 16.51283187534612652708811730119, 17.19176720882534011468393241427, 18.20004798615795870877120814506, 18.58939917155790592351451552330

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