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Properties

Label 1-469-469.81-r0-0-0
Degree $1$
Conductor $469$
Sign $-0.913 - 0.406i$
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  + (−0.327 − 0.945i)2-s + (0.580 − 0.814i)3-s + (−0.786 + 0.618i)4-s + (0.235 − 0.971i)5-s + (−0.959 − 0.281i)6-s + (0.841 + 0.540i)8-s + (−0.327 − 0.945i)9-s + (−0.995 + 0.0950i)10-s + (0.723 + 0.690i)11-s + (0.0475 + 0.998i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)15-s + (0.235 − 0.971i)16-s + (0.928 − 0.371i)17-s + (−0.786 + 0.618i)18-s + (0.981 − 0.189i)19-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (0.580 − 0.814i)3-s + (−0.786 + 0.618i)4-s + (0.235 − 0.971i)5-s + (−0.959 − 0.281i)6-s + (0.841 + 0.540i)8-s + (−0.327 − 0.945i)9-s + (−0.995 + 0.0950i)10-s + (0.723 + 0.690i)11-s + (0.0475 + 0.998i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)15-s + (0.235 − 0.971i)16-s + (0.928 − 0.371i)17-s + (−0.786 + 0.618i)18-s + (0.981 − 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $-0.913 - 0.406i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ -0.913 - 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3000736551 - 1.412609815i\)
\(L(\frac12)\) \(\approx\) \(0.3000736551 - 1.412609815i\)
\(L(1)\) \(\approx\) \(0.7308830999 - 0.8849747155i\)
\(L(1)\) \(\approx\) \(0.7308830999 - 0.8849747155i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.327 - 0.945i)T \)
3 \( 1 + (0.580 - 0.814i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
11 \( 1 + (0.723 + 0.690i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (0.928 - 0.371i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
23 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + T \)
31 \( 1 + (0.0475 - 0.998i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.995 - 0.0950i)T \)
53 \( 1 + (-0.786 + 0.618i)T \)
59 \( 1 + (0.0475 - 0.998i)T \)
61 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.235 + 0.971i)T \)
79 \( 1 + (0.0475 + 0.998i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
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

−24.42935757780311300374130399432, −23.367864135956863392761541961891, −22.50871124142777055345863723259, −21.83777412902084944149500011596, −21.02811247993649268405554320703, −19.6593448063821391797239967231, −19.07917340877254048744058415125, −18.20646942571975256849841943476, −17.25928246128894475140863130612, −16.14832308992110400335302772132, −15.84158183274786761982540673636, −14.55355416302129490098814639101, −14.18449145741136803992830499978, −13.58899508588241583487782324991, −11.76597993639994665700567147549, −10.60328788741840287073817718683, −10.01366920779622336796088449506, −9.03364308713777423257480012343, −8.27428795595293030677893918249, −7.23458819126019297550525667101, −6.19209331103615361339114061757, −5.396359514907079856708881584156, −3.95331002192144703996933217604, −3.316292277433944517101235763391, −1.61221165148142984058587173639, 0.98329975967441149899630087848, 1.61638366325741382065474392237, 2.90026558892312052090418124312, 3.900596643835341691750146257186, 5.08665192939042846994082653740, 6.39346023733400859868788166647, 7.86315828631998856729695391850, 8.27812221803482177268278051682, 9.46107212865015982209641269761, 9.85285787646189172045434343393, 11.53284124493269095673484399578, 12.12671760175757065442929706953, 12.899945043653312441560704560609, 13.66322323958070101624785153723, 14.39251141672574442653438811680, 15.858626664888458337701341431820, 16.94775487214087204259412856844, 17.79374966199697763476556748532, 18.36647076679107580749049855583, 19.40779545025149197401623484418, 20.31318430670364861621497798601, 20.4307914566183063060417666790, 21.49351680979853535761976198814, 22.66431987336349487545903500034, 23.40890161810003608083044964128

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