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Properties

Label 1-669-669.146-r1-0-0
Degree $1$
Conductor $669$
Sign $0.632 - 0.774i$
Analytic cond. $71.8940$
Root an. cond. $71.8940$
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  + (−0.985 − 0.169i)2-s + (0.942 + 0.333i)4-s + (0.858 + 0.512i)5-s + (0.660 − 0.750i)7-s + (−0.873 − 0.487i)8-s + (−0.759 − 0.649i)10-s + (0.238 − 0.971i)11-s + (−0.828 + 0.559i)13-s + (−0.778 + 0.628i)14-s + (0.778 + 0.628i)16-s + (0.721 + 0.691i)17-s + (−0.795 − 0.605i)19-s + (0.639 + 0.769i)20-s + (−0.398 + 0.916i)22-s + (0.0141 − 0.999i)23-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)2-s + (0.942 + 0.333i)4-s + (0.858 + 0.512i)5-s + (0.660 − 0.750i)7-s + (−0.873 − 0.487i)8-s + (−0.759 − 0.649i)10-s + (0.238 − 0.971i)11-s + (−0.828 + 0.559i)13-s + (−0.778 + 0.628i)14-s + (0.778 + 0.628i)16-s + (0.721 + 0.691i)17-s + (−0.795 − 0.605i)19-s + (0.639 + 0.769i)20-s + (−0.398 + 0.916i)22-s + (0.0141 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(669\)    =    \(3 \cdot 223\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(71.8940\)
Root analytic conductor: \(71.8940\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{669} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 669,\ (1:\ ),\ 0.632 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503230513 - 0.7129254413i\)
\(L(\frac12)\) \(\approx\) \(1.503230513 - 0.7129254413i\)
\(L(1)\) \(\approx\) \(0.9112142674 - 0.1366403219i\)
\(L(1)\) \(\approx\) \(0.9112142674 - 0.1366403219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 \)
good2 \( 1 + (-0.985 - 0.169i)T \)
5 \( 1 + (0.858 + 0.512i)T \)
7 \( 1 + (0.660 - 0.750i)T \)
11 \( 1 + (0.238 - 0.971i)T \)
13 \( 1 + (-0.828 + 0.559i)T \)
17 \( 1 + (0.721 + 0.691i)T \)
19 \( 1 + (-0.795 - 0.605i)T \)
23 \( 1 + (0.0141 - 0.999i)T \)
29 \( 1 + (-0.155 + 0.987i)T \)
31 \( 1 + (0.844 + 0.536i)T \)
37 \( 1 + (0.960 + 0.279i)T \)
41 \( 1 + (0.292 - 0.956i)T \)
43 \( 1 + (-0.999 + 0.0282i)T \)
47 \( 1 + (0.795 - 0.605i)T \)
53 \( 1 + (-0.0988 - 0.995i)T \)
59 \( 1 + (0.594 - 0.803i)T \)
61 \( 1 + (-0.0706 - 0.997i)T \)
67 \( 1 + (-0.759 + 0.649i)T \)
71 \( 1 + (0.398 + 0.916i)T \)
73 \( 1 + (0.702 + 0.712i)T \)
79 \( 1 + (0.974 - 0.224i)T \)
83 \( 1 + (0.999 + 0.0282i)T \)
89 \( 1 + (0.858 - 0.512i)T \)
97 \( 1 + (-0.0141 - 0.999i)T \)
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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

−22.701183582480511052866803670863, −21.5247329475356168312646645042, −20.96817845329620645521482750099, −20.24422715148412921090350930638, −19.33391934495589386900608453193, −18.376748678932485074636640064275, −17.69457500537213031934468054065, −17.16124154119533551522942800647, −16.36639866564597927595378312155, −15.08200411168774705717604721878, −14.852824476280100456176090894140, −13.57735469087036608385626995865, −12.28633836515299247259386417817, −11.91126898843455436689956969278, −10.631391286557066055749951578778, −9.633328320451709253143024181388, −9.39470760928449625985487156357, −8.12126318486897750802478162715, −7.577304492077030537304279236574, −6.25287299618371862976501341037, −5.52585533153663424620753548598, −4.60660128854248670519930402829, −2.67418535601894697884514052255, −1.98245481021565903509377426957, −0.97362976125999942035762606535, 0.62594056430562640464379572234, 1.69556631629724397619612058590, 2.621086560526009620816520024037, 3.75896224015778083890301618318, 5.16270662761648087616788880810, 6.424624369790009142052514486, 6.93751308370576612133311920080, 8.08823765247380415496071634740, 8.83359699838432067457409909155, 9.87978349240715040587409589638, 10.58707768344769129866341596305, 11.154384112831277387084755715492, 12.21219032401610041532625905877, 13.30261319452476796493909671555, 14.387080906633088871630846697324, 14.80429061806048490848099248176, 16.24965680027690741864238005217, 17.04626622709378476563728395482, 17.34671408707923369047470037048, 18.4207739264336842960949751107, 19.05618116868543345012654184205, 19.86324759819988097240196391328, 20.84584951632004075926968036593, 21.53692346669179043055803419895, 22.03838444077113703362711386637

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