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Properties

Label 1-6223-6223.1024-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.453 - 0.891i$
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  + (−0.733 + 0.680i)2-s + (0.826 − 0.563i)3-s + (0.0747 − 0.997i)4-s + (−0.5 − 0.866i)5-s + (−0.222 + 0.974i)6-s + (0.623 + 0.781i)8-s + (0.365 − 0.930i)9-s + (0.955 + 0.294i)10-s + (0.365 + 0.930i)11-s + (−0.5 − 0.866i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.365 + 0.930i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (0.826 − 0.563i)3-s + (0.0747 − 0.997i)4-s + (−0.5 − 0.866i)5-s + (−0.222 + 0.974i)6-s + (0.623 + 0.781i)8-s + (0.365 − 0.930i)9-s + (0.955 + 0.294i)10-s + (0.365 + 0.930i)11-s + (−0.5 − 0.866i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.365 + 0.930i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233485766 - 0.7563581574i\)
\(L(\frac12)\) \(\approx\) \(1.233485766 - 0.7563581574i\)
\(L(1)\) \(\approx\) \(0.9192158244 - 0.1294953073i\)
\(L(1)\) \(\approx\) \(0.9192158244 - 0.1294953073i\)

Euler product

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

−17.80418135744362178079866880420, −17.2030486624274018922068835371, −16.39767589384555279995575156464, −15.93883381689312718055779789246, −15.12788770365937902608496536069, −14.49862714621385328249383301052, −13.99638320209742471701407415632, −13.13315507333733771072829498887, −12.363256521827135789184673567645, −11.69705698966698719730174309820, −10.93681757791434888536579458863, −10.52197860884882835193471742264, −9.67091965743450932887621161204, −9.48829402000668170639816593975, −8.27267823054845782595989020694, −7.906708720591210610824102207698, −7.58471371462928194522149840261, −6.43626010773856061307113988964, −5.67617232489459075812592155749, −4.289415506229450550741640124012, −3.95399318673456727556180159268, −3.18018428730765884727396789752, −2.67153175698366032142413321009, −1.94526632145420639953544976744, −0.82082862340915030008746578949, 0.548483623837532849134897466690, 1.26415787238353078781791221604, 2.11248967291313906976668138677, 2.791838899418235914624359684957, 4.07945961858363461994732203355, 4.66450498508172160535814073499, 5.29960901081730155318872514677, 6.53038816534024829192934990577, 6.90012877060993905656728325603, 7.66558929293504779082189960593, 8.07451216647678991365277553019, 8.93619991448218768340643173725, 9.28107132604656842843505785697, 9.89907448013326701484647057800, 10.75577120166127107282264604097, 11.90315136961498220550792715053, 12.254034949408760957442773102577, 12.90198178741412535787620488312, 13.941183382637260493853113322986, 14.30510855119790614772229082249, 15.01701638934757526421397257058, 15.571088680768603971420002622214, 16.241770191214777035353539198079, 16.97123039519541464447396323433, 17.46888505850499647693656819939

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