Properties

Label 1-469-469.424-r0-0-0
Degree $1$
Conductor $469$
Sign $-0.848 + 0.528i$
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  + (0.0475 + 0.998i)2-s + (0.723 + 0.690i)3-s + (−0.995 + 0.0950i)4-s + (0.981 − 0.189i)5-s + (−0.654 + 0.755i)6-s + (−0.142 − 0.989i)8-s + (0.0475 + 0.998i)9-s + (0.235 + 0.971i)10-s + (−0.327 + 0.945i)11-s + (−0.786 − 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.981 − 0.189i)16-s + (0.580 − 0.814i)17-s + (−0.995 + 0.0950i)18-s + (−0.888 + 0.458i)19-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (0.723 + 0.690i)3-s + (−0.995 + 0.0950i)4-s + (0.981 − 0.189i)5-s + (−0.654 + 0.755i)6-s + (−0.142 − 0.989i)8-s + (0.0475 + 0.998i)9-s + (0.235 + 0.971i)10-s + (−0.327 + 0.945i)11-s + (−0.786 − 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.981 − 0.189i)16-s + (0.580 − 0.814i)17-s + (−0.995 + 0.0950i)18-s + (−0.888 + 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $-0.848 + 0.528i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ -0.848 + 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4837117818 + 1.691141702i\)
\(L(\frac12)\) \(\approx\) \(0.4837117818 + 1.691141702i\)
\(L(1)\) \(\approx\) \(0.9397521524 + 1.016878399i\)
\(L(1)\) \(\approx\) \(0.9397521524 + 1.016878399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.0475 + 0.998i)T \)
3 \( 1 + (0.723 + 0.690i)T \)
5 \( 1 + (0.981 - 0.189i)T \)
11 \( 1 + (-0.327 + 0.945i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.580 - 0.814i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.786 + 0.618i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.235 - 0.971i)T \)
53 \( 1 + (-0.995 + 0.0950i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.981 + 0.189i)T \)
79 \( 1 + (-0.786 - 0.618i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + T \)
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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

−23.55569996410340516152472423468, −22.47021264402798194352575419228, −21.436925484121621617216537477709, −21.10196364936772562640255411503, −20.049911499688167992924641774145, −19.22615536277575063098988617037, −18.66479763324235341311311112478, −17.64589744827148414680292587090, −17.21519279529007952658601530033, −15.42837431913211124530598069731, −14.39317777799686640117741414682, −13.79889547845911622613500754011, −12.94129955024991957585382689106, −12.4894080784076129347911652968, −11.04578391695219199613472934792, −10.3461716120636931896739154786, −9.31070870738733063801068327103, −8.563033865028252440831239886010, −7.61611614028929739027632532429, −6.12679897185731577708700663570, −5.39082158170876015040342120081, −3.7434026971072765331744158679, −2.88148404794511375672205283901, −2.052414275156179876998406726892, −0.923347101156688210281399466425, 1.78640125271318644509567630003, 3.0301621819174564244776452135, 4.54580124816245410733349636204, 4.88285188382705331633457868547, 6.20606574437811901817507783482, 7.15127627947392081656557015863, 8.2577989325862461679550689699, 9.11787580468278958344326876438, 9.777411247386614188163930828176, 10.5176796830278436442776217847, 12.3456479804425235693622132747, 13.201766060993198576120902064912, 14.25278306016991353857329708436, 14.47423831623543070538978937962, 15.58048325561370590802163535966, 16.493037646267472238518530345486, 17.01135807993746617985809305325, 18.14724464591885349597422000658, 18.86114523993851582204142227511, 20.0900446949579329113773184176, 21.12239042150518040281442181635, 21.52911279929516223987213051764, 22.57011926659486385622699784399, 23.36715368396061180318176711388, 24.5391014230036968116762896046

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