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Properties

Label 1-6223-6223.1515-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.975 - 0.221i$
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  + (−0.5 + 0.866i)2-s + (−0.0249 + 0.999i)3-s + (−0.5 − 0.866i)4-s + (−0.222 + 0.974i)5-s + (−0.853 − 0.521i)6-s + 8-s + (−0.998 − 0.0498i)9-s + (−0.733 − 0.680i)10-s + (−0.797 − 0.603i)11-s + (0.878 − 0.478i)12-s + (0.583 − 0.811i)13-s + (−0.969 − 0.246i)15-s + (−0.5 + 0.866i)16-s + (0.939 + 0.342i)17-s + (0.542 − 0.840i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.0249 + 0.999i)3-s + (−0.5 − 0.866i)4-s + (−0.222 + 0.974i)5-s + (−0.853 − 0.521i)6-s + 8-s + (−0.998 − 0.0498i)9-s + (−0.733 − 0.680i)10-s + (−0.797 − 0.603i)11-s + (0.878 − 0.478i)12-s + (0.583 − 0.811i)13-s + (−0.969 − 0.246i)15-s + (−0.5 + 0.866i)16-s + (0.939 + 0.342i)17-s + (0.542 − 0.840i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.975 - 0.221i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09374219143 + 0.8362582323i\)
\(L(\frac12)\) \(\approx\) \(-0.09374219143 + 0.8362582323i\)
\(L(1)\) \(\approx\) \(0.4733091077 + 0.5171685061i\)
\(L(1)\) \(\approx\) \(0.4733091077 + 0.5171685061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.0249 + 0.999i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.797 - 0.603i)T \)
13 \( 1 + (0.583 - 0.811i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.980 + 0.198i)T \)
29 \( 1 + (-0.878 - 0.478i)T \)
31 \( 1 + (0.797 - 0.603i)T \)
37 \( 1 + (0.456 + 0.889i)T \)
41 \( 1 + (-0.542 + 0.840i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.878 - 0.478i)T \)
59 \( 1 + (0.995 - 0.0995i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (0.583 - 0.811i)T \)
71 \( 1 + (0.980 - 0.198i)T \)
73 \( 1 - T \)
79 \( 1 + (0.878 - 0.478i)T \)
83 \( 1 + (-0.998 - 0.0498i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 + (-0.969 - 0.246i)T \)
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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

−17.63660332671937525313431373318, −16.7896368004246700681327145812, −16.19657016731875194866058068984, −15.640824303182894011254245807394, −14.25905069108503672798154199361, −13.85914737537280291243385361203, −13.10097116367749837008495214029, −12.58794776251599234128972080177, −12.13400978297745213285321299782, −11.4843630016863057454784053440, −10.905895308122043464397810021342, −9.918075336743818701876262407978, −9.26206487715937812754561130777, −8.70421391202315142624305137965, −7.9467369024890705167199757631, −7.54170756605377861260325311546, −6.79868517345511616901483694227, −5.66638572340073169902153665485, −5.04349967491067336570134056068, −4.23728007643918093157620597916, −3.40323135545829646023854429445, −2.52045855508641853831065331041, −1.83612008838400769526378446369, −1.16702268345907825264301065056, −0.36502968876721683880610964452, 0.76229235394748177818383157163, 2.05717452670849179972103152048, 3.17552773897410759714473530642, 3.5672430900510481830295670544, 4.47007581806464396474494072099, 5.417691624563432925431383242439, 5.93191763177415784066444777054, 6.310208886324885812822745345821, 7.6242494328137977070936665620, 7.98819690465935813971366346643, 8.4205962712943958267868040489, 9.61572010487536372096229517571, 10.05366047659423958249317854807, 10.42604679314970725669298318603, 11.21881383802328377908792290570, 11.738613766888268818480701320, 13.022992986636156100333738882505, 13.7733953403124993107200372099, 14.3104391447873801950288446507, 14.97784985682694311033284706584, 15.43318904132562785270376087917, 16.00147593404536975404098905656, 16.553619549859932011252141381257, 17.20121823527468364287760824814, 18.09533989218532469170675598593

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