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Properties

Label 1-6223-6223.2271-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.952 + 0.304i$
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  + (0.0747 − 0.997i)2-s + (0.270 − 0.962i)3-s + (−0.988 − 0.149i)4-s + (−0.988 − 0.149i)5-s + (−0.939 − 0.342i)6-s + (−0.222 + 0.974i)8-s + (−0.853 − 0.521i)9-s + (−0.222 + 0.974i)10-s + (−0.318 + 0.947i)11-s + (−0.411 + 0.911i)12-s + (−0.270 + 0.962i)13-s + (−0.411 + 0.911i)15-s + (0.955 + 0.294i)16-s + (−0.878 + 0.478i)17-s + (−0.583 + 0.811i)18-s − 19-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (0.270 − 0.962i)3-s + (−0.988 − 0.149i)4-s + (−0.988 − 0.149i)5-s + (−0.939 − 0.342i)6-s + (−0.222 + 0.974i)8-s + (−0.853 − 0.521i)9-s + (−0.222 + 0.974i)10-s + (−0.318 + 0.947i)11-s + (−0.411 + 0.911i)12-s + (−0.270 + 0.962i)13-s + (−0.411 + 0.911i)15-s + (0.955 + 0.294i)16-s + (−0.878 + 0.478i)17-s + (−0.583 + 0.811i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.952 + 0.304i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (2271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.952 + 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04700863120 - 0.3013409048i\)
\(L(\frac12)\) \(\approx\) \(-0.04700863120 - 0.3013409048i\)
\(L(1)\) \(\approx\) \(0.5215557190 - 0.4067035427i\)
\(L(1)\) \(\approx\) \(0.5215557190 - 0.4067035427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.0747 - 0.997i)T \)
3 \( 1 + (0.270 - 0.962i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (-0.318 + 0.947i)T \)
13 \( 1 + (-0.270 + 0.962i)T \)
17 \( 1 + (-0.878 + 0.478i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.698 - 0.715i)T \)
29 \( 1 + (0.661 + 0.749i)T \)
31 \( 1 + (-0.270 + 0.962i)T \)
37 \( 1 + (-0.998 + 0.0498i)T \)
41 \( 1 + (0.318 - 0.947i)T \)
43 \( 1 + (-0.542 - 0.840i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.411 + 0.911i)T \)
59 \( 1 + (-0.583 + 0.811i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.318 + 0.947i)T \)
71 \( 1 + (-0.969 + 0.246i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (0.456 + 0.889i)T \)
83 \( 1 + (0.878 - 0.478i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.0249 - 0.999i)T \)
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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

−17.733819677378369888053943335793, −17.255256481196395757274688881851, −16.36419589893087863117856213068, −15.96934105935698409414457434789, −15.47383495709121570551389713032, −14.95241157346957446641494300493, −14.39392932009950819417175714995, −13.52436185829058206635397516512, −13.099682414463495420100568552120, −12.069799713235469084338280978875, −11.32907538104279862661531751497, −10.67120851553562674813487709511, −10.00448954917345584270358025431, −9.1962469465245813935108486596, −8.52919851488195579865091483574, −7.991026017528521300087516231011, −7.59332355708482206702296932082, −6.44621982658386931095210914015, −5.91863493140935353463985391256, −4.98981813825627078432940280756, −4.546788595034803737364760295278, −3.73189273398758951215442242242, −3.25001790081607519940658262788, −2.38067050645524236434127596954, −0.56043271201656509839125178830, 0.127141917824008800177794459998, 1.28960268536227411509152114192, 2.079947230690299547150829649957, 2.49051724116851894277701151058, 3.5532393557864330456312948443, 4.20221256383550110036094919664, 4.7443147607922106600566067446, 5.71221573163746260353070036050, 6.890198073109130621235161167646, 7.07770549074904895809099844391, 8.179875209212407455794744516487, 8.72607541303414794002423042138, 9.033006764434832846824229377833, 10.33931320732059617284339358046, 10.67169705575632498185612860556, 11.6379538745742498467318312761, 12.19560841066692402432280025178, 12.45654978453941043762195548932, 13.139104323709413573376222535336, 13.91756581419600582403841165354, 14.53457695324426349733781621004, 15.094137783856042900585179777298, 15.91311032071674646151294013688, 16.9402433214665576185705674518, 17.47009474725330719027532569753

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