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Properties

Label 1-469-469.464-r0-0-0
Degree $1$
Conductor $469$
Sign $0.0114 - 0.999i$
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  + (−0.786 + 0.618i)2-s + (−0.327 − 0.945i)3-s + (0.235 − 0.971i)4-s + (−0.888 − 0.458i)5-s + (0.841 + 0.540i)6-s + (0.415 + 0.909i)8-s + (−0.786 + 0.618i)9-s + (0.981 − 0.189i)10-s + (0.0475 + 0.998i)11-s + (−0.995 + 0.0950i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (0.723 − 0.690i)17-s + (0.235 − 0.971i)18-s + (0.928 − 0.371i)19-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (−0.327 − 0.945i)3-s + (0.235 − 0.971i)4-s + (−0.888 − 0.458i)5-s + (0.841 + 0.540i)6-s + (0.415 + 0.909i)8-s + (−0.786 + 0.618i)9-s + (0.981 − 0.189i)10-s + (0.0475 + 0.998i)11-s + (−0.995 + 0.0950i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (0.723 − 0.690i)17-s + (0.235 − 0.971i)18-s + (0.928 − 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0114 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0114 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.0114 - 0.999i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.0114 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4116153154 - 0.4163659924i\)
\(L(\frac12)\) \(\approx\) \(0.4116153154 - 0.4163659924i\)
\(L(1)\) \(\approx\) \(0.5678518991 - 0.1435297391i\)
\(L(1)\) \(\approx\) \(0.5678518991 - 0.1435297391i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
3 \( 1 + (-0.327 - 0.945i)T \)
5 \( 1 + (-0.888 - 0.458i)T \)
11 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.723 - 0.690i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.981 + 0.189i)T \)
53 \( 1 + (0.235 - 0.971i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
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

−23.86536377943652816145726153035, −23.09495048098150809648963020007, −22.049612939248393722083134796475, −21.52686725771900824278328688329, −20.65587100634357463845076063134, −19.8019493421450603389597022764, −18.903452769849815661484356033404, −18.35860586806915510998361378753, −17.00375544900798122081743128217, −16.45620722956213029799735306259, −15.76472063639189292647031143011, −14.748285656831939251078614269047, −13.65855633688406822319216848617, −12.12003662328666350247826321283, −11.627455548307033096522513959650, −10.80214754154650480129331926168, −10.17294226581815110090320066848, −8.96705806903928332012456230686, −8.38151429275629268229946713539, −7.191361159295453444612545254197, −6.089043200556873536564507747240, −4.60461966090133071200332251147, −3.52847891719168159068753958179, −3.09294044280669760287316092282, −1.15444015612484563217987696168, 0.548932197741442188918551980523, 1.55846482844634194418761656655, 3.08656644779699754093352979452, 4.923076406240976774963474574418, 5.54165440428951248611788471104, 7.00140201393906722256417343212, 7.3910227256528376849109501721, 8.26496224627018202240529653690, 9.16993229202129717783960960448, 10.387076981029428307161629812053, 11.39963670095865865493365655527, 12.164842629647871186432617017335, 13.11629147117060531056162115636, 14.21633089782976090826752366375, 15.25499523349501613417580382982, 15.98907281197285933727971593962, 16.89362148051626667168689999018, 17.70481045174850872512003072750, 18.39769109095300756422115092836, 19.19811038393255716082463910900, 20.10076691878403771522990232498, 20.504907978046436229374195234031, 22.45059678914197632125728948389, 23.3099911105050575862656772877, 23.49698588772145873425186742144

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