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Properties

Label 1-6223-6223.496-r1-0-0
Degree $1$
Conductor $6223$
Sign $0.989 - 0.141i$
Analytic cond. $668.754$
Root an. cond. $668.754$
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  + (−0.900 − 0.433i)2-s + (−0.270 + 0.962i)3-s + (0.623 + 0.781i)4-s + (−0.0747 + 0.997i)5-s + (0.661 − 0.749i)6-s + (−0.222 − 0.974i)8-s + (−0.853 − 0.521i)9-s + (0.5 − 0.866i)10-s + (0.980 − 0.198i)11-s + (−0.921 + 0.388i)12-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (−0.222 + 0.974i)16-s + (0.318 − 0.947i)17-s + (0.542 + 0.840i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.270 + 0.962i)3-s + (0.623 + 0.781i)4-s + (−0.0747 + 0.997i)5-s + (0.661 − 0.749i)6-s + (−0.222 − 0.974i)8-s + (−0.853 − 0.521i)9-s + (0.5 − 0.866i)10-s + (0.980 − 0.198i)11-s + (−0.921 + 0.388i)12-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (−0.222 + 0.974i)16-s + (0.318 − 0.947i)17-s + (0.542 + 0.840i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(668.754\)
Root analytic conductor: \(668.754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (1:\ ),\ 0.989 - 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8943594749 - 0.06372215594i\)
\(L(\frac12)\) \(\approx\) \(0.8943594749 - 0.06372215594i\)
\(L(1)\) \(\approx\) \(0.6178622437 + 0.2149001295i\)
\(L(1)\) \(\approx\) \(0.6178622437 + 0.2149001295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (-0.270 + 0.962i)T \)
5 \( 1 + (-0.0747 + 0.997i)T \)
11 \( 1 + (0.980 - 0.198i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.318 - 0.947i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.456 + 0.889i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.921 + 0.388i)T \)
37 \( 1 + (-0.583 + 0.811i)T \)
41 \( 1 + (-0.270 - 0.962i)T \)
43 \( 1 + (0.921 + 0.388i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.921 + 0.388i)T \)
59 \( 1 + (0.318 - 0.947i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.995 - 0.0995i)T \)
71 \( 1 + (0.542 - 0.840i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (0.878 + 0.478i)T \)
83 \( 1 + (0.0249 - 0.999i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
97 \( 1 + (-0.698 - 0.715i)T \)
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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

−17.45847365739250535404399145319, −16.97850124302592896524646967497, −16.538485717541518208130784344203, −15.731494483471185977904539704490, −14.864348850754882772070750709083, −14.440912161365037972840606590968, −13.48618501392646597939022014874, −12.83259773764297301673597099890, −12.13654586784237702706074358881, −11.72188741675540357055360626633, −10.8938944255359434864624217913, −10.108100927535268094869077753073, −9.366581693080972527055198513705, −8.635390366687223896848868439236, −8.28682071858048778638409340581, −7.38919181844995951507876109789, −6.98719348022123570940563158719, −6.14444525239024245716943460953, −5.493877723848479506057844865658, −4.92714511695066101689756961050, −3.879400030383847421688896565296, −2.57356247489044623597782005718, −2.006842184462366446758059604535, −0.96413608862990328548772109983, −0.769098179102784390686687436204, 0.244537291308706494320685140664, 1.291741546094924154684605118659, 2.21445298406638344536280914762, 3.22777204789063784578129902116, 3.41789237510277465936491262186, 4.31271698598804998174034681503, 5.269848996567916713726860251454, 6.138895769003321051911349439416, 6.9393138521858403540641764463, 7.325725494634309848971713069144, 8.314485275235085202145441687769, 9.14387417292276864106921371948, 9.63333374787879460992527299565, 10.04557244273702555961282741807, 10.96006173810280172885509617825, 11.266675730414498568957848255780, 12.02591988474132546159269342087, 12.35066638751037620782649162780, 13.9453522113787841992973110925, 14.18052618378200645280545099379, 15.0002905163292050458501795969, 15.68318268804872869367574398558, 16.328909528296456927355006056810, 16.81024873375695901676966766733, 17.55529714170833266362536974570

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