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Properties

Label 1-469-469.102-r0-0-0
Degree $1$
Conductor $469$
Sign $0.804 - 0.593i$
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  + (0.841 − 0.540i)2-s + (0.723 + 0.690i)3-s + (0.415 − 0.909i)4-s + (0.981 − 0.189i)5-s + (0.981 + 0.189i)6-s + (−0.142 − 0.989i)8-s + (0.0475 + 0.998i)9-s + (0.723 − 0.690i)10-s + (−0.654 − 0.755i)11-s + (0.928 − 0.371i)12-s + (0.928 − 0.371i)13-s + (0.841 + 0.540i)15-s + (−0.654 − 0.755i)16-s + (−0.995 − 0.0950i)17-s + (0.580 + 0.814i)18-s + (0.841 + 0.540i)19-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.723 + 0.690i)3-s + (0.415 − 0.909i)4-s + (0.981 − 0.189i)5-s + (0.981 + 0.189i)6-s + (−0.142 − 0.989i)8-s + (0.0475 + 0.998i)9-s + (0.723 − 0.690i)10-s + (−0.654 − 0.755i)11-s + (0.928 − 0.371i)12-s + (0.928 − 0.371i)13-s + (0.841 + 0.540i)15-s + (−0.654 − 0.755i)16-s + (−0.995 − 0.0950i)17-s + (0.580 + 0.814i)18-s + (0.841 + 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.804 - 0.593i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.804 - 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.997514208 - 0.9857107686i\)
\(L(\frac12)\) \(\approx\) \(2.997514208 - 0.9857107686i\)
\(L(1)\) \(\approx\) \(2.232205296 - 0.4969276409i\)
\(L(1)\) \(\approx\) \(2.232205296 - 0.4969276409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (0.723 + 0.690i)T \)
5 \( 1 + (0.981 - 0.189i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (0.928 - 0.371i)T \)
17 \( 1 + (-0.995 - 0.0950i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.723 + 0.690i)T \)
53 \( 1 + (-0.995 + 0.0950i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (0.580 + 0.814i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (0.928 - 0.371i)T \)
83 \( 1 + (-0.327 + 0.945i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
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

−23.99649728114288858005902549882, −23.30056749306419456792132325961, −22.25590107682675628271517531410, −21.48152038627615744625456583112, −20.51553872675236383577238588680, −20.127075824514682422549482828213, −18.51721922953489841746015467638, −17.971889712253456076031156605095, −17.18778930548375200630972239703, −15.85738591758248212331984617290, −15.202962911426264076180725499035, −14.13098639559517448181936084508, −13.60401891210049477838258900678, −13.00378726590985278958849965296, −12.06158412303847015836044563947, −10.89123233721356410760244713165, −9.55150973190662265731132511173, −8.63911791766783048796421381788, −7.62774909310343562117167381148, −6.72122543102017258927090603043, −6.057092006661564556705788241655, −4.87702118942727232370982747852, −3.65793538714702221829980012999, −2.51644586815501783469009348449, −1.81791359323504969097770863684, 1.474965772538177521908955053354, 2.55682022053663235368936985788, 3.39229053835673919888973767923, 4.45908563190151443083542560068, 5.51676783479911344242953168042, 6.131772531714417194589800419691, 7.76440496473066140668878029716, 8.951239197583324289991195729029, 9.73225248262381684688190589027, 10.618244394034048806955661292419, 11.26173798700987955461246571806, 12.76568780911831805083372803440, 13.56867939841738591903446935100, 13.89622435161769124893073838913, 14.96723150013509533198477357580, 15.87925994766233535661945678164, 16.48509933329182730104411634010, 18.07853459659389784647657883431, 18.735045771715632410201487847422, 20.041039753552344056097653921794, 20.48883910608382898102254607771, 21.21294121615836423552687207549, 21.97836414142552924272455978990, 22.49384116426447676771014527623, 23.85036935214826347396664712798

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