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Properties

Label 1-6223-6223.3778-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.859 + 0.511i$
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 + (0.955 + 0.294i)3-s + (−0.5 + 0.866i)4-s + (0.826 + 0.563i)5-s + (−0.222 − 0.974i)6-s + 8-s + (0.826 + 0.563i)9-s + (0.0747 − 0.997i)10-s + (0.826 − 0.563i)11-s + (−0.733 + 0.680i)12-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.0747 − 0.997i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.955 + 0.294i)3-s + (−0.5 + 0.866i)4-s + (0.826 + 0.563i)5-s + (−0.222 − 0.974i)6-s + 8-s + (0.826 + 0.563i)9-s + (0.0747 − 0.997i)10-s + (0.826 − 0.563i)11-s + (−0.733 + 0.680i)12-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.0747 − 0.997i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (3778, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.487010187 + 0.6849075488i\)
\(L(\frac12)\) \(\approx\) \(2.487010187 + 0.6849075488i\)
\(L(1)\) \(\approx\) \(1.395363071 - 0.04600962900i\)
\(L(1)\) \(\approx\) \(1.395363071 - 0.04600962900i\)

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 \( 1 + (0.955 + 0.294i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.955 + 0.294i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.0747 + 0.997i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 - T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (-0.365 - 0.930i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.733 + 0.680i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−17.70628512950597518734003094484, −16.96783763111743107485394762344, −16.385983368945839589371488367122, −15.55108863308452983245501046844, −14.91434856814681522521704001743, −14.40424345701027194183933011585, −13.93974660074735647251119441289, −13.01675411232375771067471031617, −12.759211889074571094684416005132, −11.8468238638578168337537854106, −10.57586571765356664360127247804, −9.977475118029455424006558993944, −9.44483439119491331611384937424, −8.98597073950645196468485909878, −8.2212805845437345920760936130, −7.65080847781350010972561309900, −6.99932304373376181410939901741, −6.18929015554401069116711705389, −5.664723301453914968753890161191, −4.66120074977727316021925321603, −4.18179182192601516353823494686, −3.020791059671475738762982975106, −2.09417110280195936608654072507, −1.47603881763361135560556161474, −0.68994917008436711889387421000, 1.12682764368543743000407894980, 1.77087346083904865650424110779, 2.502659261876194489488236604820, 3.096731728674019403917813994082, 3.79270446093974565167310284900, 4.46527314519836012568554134902, 5.41338381756318400280192552703, 6.39176247163682883832200041217, 7.26283520087871708692514575028, 7.77770387340052256168614652156, 8.64364910653666303539350439646, 9.333453198488939013851241886886, 9.68670308528418417699487354533, 10.227387303868941842424055419624, 10.94939500678956240796805482477, 11.90277798566261220178846404939, 12.17142091320078405216283716860, 13.368072964683964204287648842577, 13.89659990455963029123195633420, 14.120106620502177158104301238456, 14.840143368706027772012834470875, 15.88406602213888968769186815349, 16.620105079959561699293987777932, 16.99696572242927388869768811559, 17.97946890372065705708668297460

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