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Properties

Label 1-416-416.3-r1-0-0
Degree $1$
Conductor $416$
Sign $0.560 - 0.827i$
Analytic cond. $44.7054$
Root an. cond. $44.7054$
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  + (0.965 − 0.258i)3-s + (−0.707 + 0.707i)5-s + (0.866 − 0.5i)7-s + (0.866 − 0.5i)9-s + (−0.258 − 0.965i)11-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.707 − 0.707i)21-s + (0.866 + 0.5i)23-s − i·25-s + (0.707 − 0.707i)27-s + (−0.965 + 0.258i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)3-s + (−0.707 + 0.707i)5-s + (0.866 − 0.5i)7-s + (0.866 − 0.5i)9-s + (−0.258 − 0.965i)11-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.707 − 0.707i)21-s + (0.866 + 0.5i)23-s − i·25-s + (0.707 − 0.707i)27-s + (−0.965 + 0.258i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.560 - 0.827i$
Analytic conductor: \(44.7054\)
Root analytic conductor: \(44.7054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 416,\ (1:\ ),\ 0.560 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.458507996 - 1.303983584i\)
\(L(\frac12)\) \(\approx\) \(2.458507996 - 1.303983584i\)
\(L(1)\) \(\approx\) \(1.485100251 - 0.2670042864i\)
\(L(1)\) \(\approx\) \(1.485100251 - 0.2670042864i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.965 + 0.258i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (0.258 - 0.965i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−24.37878923748669187477163869302, −23.41865925687024718907647857896, −22.44748322258984277115158918112, −21.23800686912107793588059416285, −20.47748859642815593977595577031, −20.287749975158590067170721519660, −18.87686076796977618786906874094, −18.43014803201427122530910052782, −17.04471805488427713865284061137, −16.15029787661478681770445440979, −15.17746313474175050699388923566, −14.75758541905933237944734693563, −13.6383261248381243873263188795, −12.57105367147744311100695471615, −11.894989102891551323550419169501, −10.68678060473450961318702323093, −9.51044470555526213642198850414, −8.79964138393189330538015389365, −7.828424906587970514756637440806, −7.31670416324467143423055550843, −5.34209104117110135718792233455, −4.639709651531171741931949610836, −3.6255208791108578943073734049, −2.35023364402582794134854578795, −1.25994160400902931929584733224, 0.72450638007854219134665926525, 2.07817423839398731770323464124, 3.32491232501701560436412047578, 3.931192632654218897667721411812, 5.36123360687540493930326394829, 6.87760998274553685017963674683, 7.58498777441427595359029082468, 8.31420367432436505557746589033, 9.27679086812631039233300013457, 10.72243868573470831410456386747, 11.1475938778970959630266183399, 12.44681155279851816967819065936, 13.50030711553350815539516274086, 14.26334450282120911749415596149, 14.97711200571832424573053275554, 15.732825671334271636220023980610, 16.95159260857166460254496241387, 18.06347027021248441657571256346, 18.84740540906343951828120203224, 19.51143421965660950465928641294, 20.3347255809388468228868256818, 21.268367938307372428218373479182, 21.98537235702983490102588887642, 23.38348485795278392781941200744, 23.90439327156844961825846571992

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