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Properties

Label 1-469-469.394-r0-0-0
Degree $1$
Conductor $469$
Sign $0.791 + 0.611i$
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  + (0.235 + 0.971i)2-s + (−0.786 − 0.618i)3-s + (−0.888 + 0.458i)4-s + (0.580 − 0.814i)5-s + (0.415 − 0.909i)6-s + (−0.654 − 0.755i)8-s + (0.235 + 0.971i)9-s + (0.928 + 0.371i)10-s + (−0.995 − 0.0950i)11-s + (0.981 + 0.189i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (0.0475 + 0.998i)17-s + (−0.888 + 0.458i)18-s + (0.723 + 0.690i)19-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)2-s + (−0.786 − 0.618i)3-s + (−0.888 + 0.458i)4-s + (0.580 − 0.814i)5-s + (0.415 − 0.909i)6-s + (−0.654 − 0.755i)8-s + (0.235 + 0.971i)9-s + (0.928 + 0.371i)10-s + (−0.995 − 0.0950i)11-s + (0.981 + 0.189i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (0.0475 + 0.998i)17-s + (−0.888 + 0.458i)18-s + (0.723 + 0.690i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.791 + 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9841784691 + 0.3358366917i\)
\(L(\frac12)\) \(\approx\) \(0.9841784691 + 0.3358366917i\)
\(L(1)\) \(\approx\) \(0.8778309542 + 0.2314801830i\)
\(L(1)\) \(\approx\) \(0.8778309542 + 0.2314801830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.235 + 0.971i)T \)
3 \( 1 + (-0.786 - 0.618i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
11 \( 1 + (-0.995 - 0.0950i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.928 - 0.371i)T \)
53 \( 1 + (-0.888 + 0.458i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
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

−23.27976456616321979248099164330, −22.66988514430197789561487592750, −22.13474029244225094658036167883, −21.19648074415758741093680090121, −20.752802220161289220404599899415, −19.57969525353623733391312914655, −18.523782843963270535222866213797, −17.81286793823898182845606729128, −17.34716056643313667090654204792, −15.80890058042060695624848828492, −15.1289762871541093658360707605, −14.07757152407727335954820283368, −13.21968542954029706912300649967, −12.22409138992576119896913299308, −11.308562900557147342204126654353, −10.592303965981660946129855117491, −9.925320335647073726685120681383, −9.22970768385592571008512528393, −7.58568444179108324982388628185, −6.339458875337960208260328718858, −5.215007534807392981463284793771, −4.801770687057449609591484948644, −3.13463940204030974371869518694, −2.67394693267182790727737840733, −0.861997555858903742179568927, 0.90591100932826523131939797933, 2.39410652183440817091852066564, 4.2554128206343707863019423523, 5.16443388567334080924405769847, 5.774610383511076969374407339398, 6.734165994120767090813789932095, 7.71694639341000335104598371315, 8.53423431592573336582697995941, 9.670148212837969117016282448019, 10.67471188017777504199460185591, 12.27846179311589875588057701526, 12.53225265550615142699873333324, 13.583461123788879772900136822828, 14.18890520007309851924018952747, 15.59689453830578521072058498387, 16.327466076842784778660810555491, 17.11509557844837311712661088232, 17.59502575280124977282524408383, 18.58128481602385336457799967187, 19.35142266617972470859208689342, 20.954995540620895773172153479793, 21.52204603271127472872971614216, 22.48858291473458018317641713835, 23.35023352702796233665194860292, 24.0459105249291301019350296421

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