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Properties

Label 1-6223-6223.20-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.213 - 0.977i$
Analytic cond. $28.8994$
Root an. cond. $28.8994$
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  + (0.623 + 0.781i)2-s + (0.0747 − 0.997i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + (0.826 − 0.563i)6-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (−0.900 − 0.433i)10-s + (0.365 − 0.930i)11-s + (0.955 + 0.294i)12-s + (0.988 − 0.149i)13-s + (0.365 + 0.930i)15-s + (−0.900 − 0.433i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)18-s − 19-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.0747 − 0.997i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + (0.826 − 0.563i)6-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (−0.900 − 0.433i)10-s + (0.365 − 0.930i)11-s + (0.955 + 0.294i)12-s + (0.988 − 0.149i)13-s + (0.365 + 0.930i)15-s + (−0.900 − 0.433i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.213 - 0.977i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.213 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6643128055 - 0.8248745056i\)
\(L(\frac12)\) \(\approx\) \(0.6643128055 - 0.8248745056i\)
\(L(1)\) \(\approx\) \(1.103680975 + 0.07380522428i\)
\(L(1)\) \(\approx\) \(1.103680975 + 0.07380522428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (0.0747 - 0.997i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
19 \( 1 - T \)
23 \( 1 + (0.733 + 0.680i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (-0.0747 - 0.997i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.955 + 0.294i)T \)
59 \( 1 + (0.0747 + 0.997i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.733 - 0.680i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.988 - 0.149i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−17.76072630967247007470651167260, −17.16320479953916656894893032449, −16.18758986117218708709502665201, −15.88233696810327532919020699381, −15.0474285837511029610593742223, −14.6436209577405526622128672927, −14.117249908627498095191815935118, −12.96703062142869076373840357720, −12.59796569598694353471808894210, −11.910315040617338367915645936232, −11.18841550166518074058410559277, −10.71211001021437003564614418264, −10.08065038100544392794325441770, −9.33853286972137716423653227266, −8.57152324938600894036038433576, −8.25020434397398761150485967010, −6.8722498386585872427140698412, −6.2627888976339450659322629490, −5.30281135877981641906233391767, −4.550989156120972332824337491078, −4.364382954840554959703556080134, −3.39789300097880150645918576138, −3.107625309634011083519958605609, −1.83793054049292726576525743181, −1.1100657074055028836610450289, 0.23379753507955939819800075736, 1.19479588278161314283922850973, 2.46692819920743640955813931699, 3.241901604638963060622121036677, 3.606918559407318823133200273771, 4.541513363128900990101635631802, 5.52832915856719855055787067486, 6.105260504382534342956332084786, 6.76147460629503126698354956201, 7.25017644162266768145042462977, 8.09363498926145868828587447412, 8.43432063598571099291807763904, 9.02729048029253469006439274376, 10.35850398323289657804791464841, 11.30907439003751601162980392797, 11.689523962324237044011844727822, 12.20603804254794869499301058547, 13.10957728825925888496994134002, 13.62499230172860515455639014252, 14.07595368250285717245346991311, 14.87060069522982602003327324953, 15.42818097024822050749017131026, 16.04128122744686872624762027029, 16.896426969630966970337427440629, 17.22640402303749480862854944213

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