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Properties

Label 1-1421-1421.821-r0-0-0
Degree $1$
Conductor $1421$
Sign $0.954 - 0.297i$
Analytic cond. $6.59909$
Root an. cond. $6.59909$
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  + (0.5 + 0.866i)2-s + (−0.955 − 0.294i)3-s + (−0.5 + 0.866i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s − 8-s + (0.826 + 0.563i)9-s + (0.733 + 0.680i)10-s + (−0.826 + 0.563i)11-s + (0.733 − 0.680i)12-s + (−0.900 − 0.433i)13-s − 15-s + (−0.5 − 0.866i)16-s + (−0.955 − 0.294i)17-s + (−0.0747 + 0.997i)18-s + (0.733 − 0.680i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.955 − 0.294i)3-s + (−0.5 + 0.866i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s − 8-s + (0.826 + 0.563i)9-s + (0.733 + 0.680i)10-s + (−0.826 + 0.563i)11-s + (0.733 − 0.680i)12-s + (−0.900 − 0.433i)13-s − 15-s + (−0.5 − 0.866i)16-s + (−0.955 − 0.294i)17-s + (−0.0747 + 0.997i)18-s + (0.733 − 0.680i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1421\)    =    \(7^{2} \cdot 29\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(6.59909\)
Root analytic conductor: \(6.59909\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1421} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1421,\ (0:\ ),\ 0.954 - 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063562217 - 0.1616623880i\)
\(L(\frac12)\) \(\approx\) \(1.063562217 - 0.1616623880i\)
\(L(1)\) \(\approx\) \(0.9235732703 + 0.2551610305i\)
\(L(1)\) \(\approx\) \(0.9235732703 + 0.2551610305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (0.733 - 0.680i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (-0.365 - 0.930i)T \)
37 \( 1 + (-0.955 + 0.294i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.826 - 0.563i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.0747 - 0.997i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
97 \( 1 + (0.222 + 0.974i)T \)
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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

−21.076369382096870462070021896162, −20.33594708050795409550714483860, −19.1411648151900456233430865775, −18.6043112459116757276804624450, −17.79526453354780976209946063973, −17.29102508865737434899831757636, −16.25002326608441758195582646691, −15.47178057176436625643251071076, −14.46434161165758188922860298343, −13.9231272776172572541434454632, −12.82158742547128826099656838883, −12.57031815868911958984209864911, −11.37496492639159070681338853364, −10.87798181306169378471475133001, −10.18216862758456796264175078933, −9.5652450345963285815634062654, −8.699449991563481323688321496488, −7.10022526184854751811730171508, −6.365618532612232969731449206648, −5.417738627750970882553805156685, −5.08864597806879371743441919159, −4.03513107757619582586148440221, −2.94811936790410896401764199839, −2.09920468896235246035109902987, −1.04615667565939827402832489212, 0.4340853642174016742803380171, 2.0024426808583078661487351018, 2.894947033625154669202743625970, 4.45878350249546578413200209013, 5.124240461741740297908325653941, 5.48808014088347007447459485512, 6.526576181480578622303963574658, 7.1820792758393269320493725690, 7.8593143069214788980616669780, 9.14700411570729168432913776009, 9.75478481297385447944393800254, 10.759582269648828927860218146636, 11.72165127571617667124477950001, 12.602143460192426000809292476539, 13.16442091054279524778641751199, 13.61169410575157516225542549811, 14.74950445060320280485054286291, 15.54966143214379444180236241410, 16.1833476444681936413439061488, 17.08409273148509414370721147952, 17.66313314522977168134456315822, 17.909435121503118284927661885486, 18.897629301014602357788098558917, 20.17667804562570091764534264917, 21.0220696801564451805630734355

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