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Properties

Label 1-469-469.111-r0-0-0
Degree $1$
Conductor $469$
Sign $-0.607 - 0.794i$
Analytic cond. $2.17802$
Root an. cond. $2.17802$
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  + (0.888 − 0.458i)2-s + (−0.959 − 0.281i)3-s + (0.580 − 0.814i)4-s + (−0.654 + 0.755i)5-s + (−0.981 + 0.189i)6-s + (0.142 − 0.989i)8-s + (0.841 + 0.540i)9-s + (−0.235 + 0.971i)10-s + (−0.981 − 0.189i)11-s + (−0.786 + 0.618i)12-s + (0.928 + 0.371i)13-s + (0.841 − 0.540i)15-s + (−0.327 − 0.945i)16-s + (−0.580 − 0.814i)17-s + (0.995 + 0.0950i)18-s + (−0.0475 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.888 − 0.458i)2-s + (−0.959 − 0.281i)3-s + (0.580 − 0.814i)4-s + (−0.654 + 0.755i)5-s + (−0.981 + 0.189i)6-s + (0.142 − 0.989i)8-s + (0.841 + 0.540i)9-s + (−0.235 + 0.971i)10-s + (−0.981 − 0.189i)11-s + (−0.786 + 0.618i)12-s + (0.928 + 0.371i)13-s + (0.841 − 0.540i)15-s + (−0.327 − 0.945i)16-s + (−0.580 − 0.814i)17-s + (0.995 + 0.0950i)18-s + (−0.0475 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $-0.607 - 0.794i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ -0.607 - 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5053002116 - 1.022511037i\)
\(L(\frac12)\) \(\approx\) \(0.5053002116 - 1.022511037i\)
\(L(1)\) \(\approx\) \(0.9548583255 - 0.5066918165i\)
\(L(1)\) \(\approx\) \(0.9548583255 - 0.5066918165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.888 - 0.458i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (-0.981 - 0.189i)T \)
13 \( 1 + (0.928 + 0.371i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.995 + 0.0950i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (-0.235 - 0.971i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (-0.981 + 0.189i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (0.327 + 0.945i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
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

−23.80403502762924974395735380153, −23.368721465719396304584186875898, −22.761619061059651641543183049027, −21.69522754933975278112521171308, −20.90427061262076763965191718478, −20.38596855514501788463215598754, −19.00618950890151463935697910543, −17.828300333644444763148852940582, −17.06238377429097840788640665763, −16.17724778657211635897104512018, −15.651186938966883759863669638625, −14.99178473323875858954108760208, −13.456388365561527154581005472649, −12.78310877781329146239412176131, −12.13385427849428104271951725860, −11.135379072705589919104686104486, −10.445298972349600949831122681489, −8.81057303502360457150158372422, −7.91268554960291834694448075943, −6.918803359809500914395691007056, −5.76034353174368758982752866151, −5.18103792018661486211496646074, −4.189915948123998499917595475906, −3.39130011479815061278227274282, −1.51299977636472630272590059346, 0.53969501697985408896633685856, 2.18377530294089599880808156734, 3.21253635233488563265773885731, 4.45383375417240489558271884260, 5.183174896172553478734943388027, 6.48775814194126442607237463906, 6.85443116419357786957298836826, 8.134942558833045628908366287047, 9.84592194367112147656921286031, 10.83986489238126538572226803218, 11.29582720647797176646605243757, 11.96647088787961714473159670815, 13.25020211157217230480516392008, 13.56017251508808761565639298016, 15.03187142696010025205876045448, 15.6811232927769392753244666557, 16.36119694123943503692152130646, 17.7815519805283774993324938143, 18.71101312127583245404423284214, 19.04400059422006740551494809856, 20.349365401084273651844101262543, 21.18131804015098942812459008792, 22.10843095066493578152691475843, 22.76013843567463237556976949136, 23.3880625079815863457576460828

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