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Properties

Label 1-223-223.28-r0-0-0
Degree $1$
Conductor $223$
Sign $-0.305 - 0.952i$
Analytic cond. $1.03560$
Root an. cond. $1.03560$
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  + (0.524 + 0.851i)2-s + (−0.911 − 0.411i)3-s + (−0.450 + 0.892i)4-s + (−0.996 − 0.0848i)5-s + (−0.127 − 0.991i)6-s + (0.372 + 0.927i)7-s + (−0.996 + 0.0848i)8-s + (0.660 + 0.750i)9-s + (−0.450 − 0.892i)10-s + (−0.127 − 0.991i)11-s + (0.778 − 0.628i)12-s + (−0.911 + 0.411i)13-s + (−0.594 + 0.803i)14-s + (0.873 + 0.487i)15-s + (−0.594 − 0.803i)16-s + (−0.127 − 0.991i)17-s + ⋯
L(s)  = 1  + (0.524 + 0.851i)2-s + (−0.911 − 0.411i)3-s + (−0.450 + 0.892i)4-s + (−0.996 − 0.0848i)5-s + (−0.127 − 0.991i)6-s + (0.372 + 0.927i)7-s + (−0.996 + 0.0848i)8-s + (0.660 + 0.750i)9-s + (−0.450 − 0.892i)10-s + (−0.127 − 0.991i)11-s + (0.778 − 0.628i)12-s + (−0.911 + 0.411i)13-s + (−0.594 + 0.803i)14-s + (0.873 + 0.487i)15-s + (−0.594 − 0.803i)16-s + (−0.127 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(223\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(1.03560\)
Root analytic conductor: \(1.03560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{223} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 223,\ (0:\ ),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03424742865 - 0.04692951972i\)
\(L(\frac12)\) \(\approx\) \(0.03424742865 - 0.04692951972i\)
\(L(1)\) \(\approx\) \(0.5526853841 + 0.2244700788i\)
\(L(1)\) \(\approx\) \(0.5526853841 + 0.2244700788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad223 \( 1 \)
good2 \( 1 + (0.524 + 0.851i)T \)
3 \( 1 + (-0.911 - 0.411i)T \)
5 \( 1 + (-0.996 - 0.0848i)T \)
7 \( 1 + (0.372 + 0.927i)T \)
11 \( 1 + (-0.127 - 0.991i)T \)
13 \( 1 + (-0.911 + 0.411i)T \)
17 \( 1 + (-0.127 - 0.991i)T \)
19 \( 1 + (-0.721 - 0.691i)T \)
23 \( 1 + (-0.996 - 0.0848i)T \)
29 \( 1 + (-0.594 - 0.803i)T \)
31 \( 1 + (-0.967 - 0.251i)T \)
37 \( 1 + (-0.127 + 0.991i)T \)
41 \( 1 + (0.210 - 0.977i)T \)
43 \( 1 + (0.985 - 0.169i)T \)
47 \( 1 + (-0.721 + 0.691i)T \)
53 \( 1 + (-0.828 + 0.559i)T \)
59 \( 1 + (0.778 + 0.628i)T \)
61 \( 1 + (-0.911 + 0.411i)T \)
67 \( 1 + (-0.450 + 0.892i)T \)
71 \( 1 + (0.778 + 0.628i)T \)
73 \( 1 + (0.0424 - 0.999i)T \)
79 \( 1 + (0.210 - 0.977i)T \)
83 \( 1 + (0.985 + 0.169i)T \)
89 \( 1 + (-0.996 + 0.0848i)T \)
97 \( 1 + (-0.996 + 0.0848i)T \)
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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

−27.22269522342802876955301888514, −26.21021279394698083128859420518, −24.27144339924900676196868398163, −23.61252606436184080141510281467, −22.98146572202880847599425335432, −22.22436167189524028813592326166, −21.19501257301375458526328715443, −20.18044727467260432053072734431, −19.61074682340562744963370339539, −18.26067804408694030060558010204, −17.41202296801820650992291003572, −16.29049607251395498935515588668, −15.0309606055242712250089755654, −14.55361581101866035887293330023, −12.69496384378583439625387159450, −12.38193998381389913972352717692, −11.117826389798647302846365381983, −10.566355345747575330716032522201, −9.65235475220955060806959957913, −7.891605310141999499005709494668, −6.708464540522863129271461072999, −5.26356002605501438970648509460, −4.306715885395680971634406275566, −3.69589442537302091257060784385, −1.72077963065749828087903754462, 0.03934758939018773510645279308, 2.57151604465234518862689194933, 4.24085540150002333687417666214, 5.138776429918033301935724335539, 6.102041966044927682896754627423, 7.23700811330821594394382912997, 8.05779426150454546267526218770, 9.170488420940560669453178079071, 11.153108034806590578982205849568, 11.84096957233558137020764235934, 12.55334943745162356753771317804, 13.72391141556272343161388355361, 14.91650876245115425202637185854, 15.84001491232406445591543374932, 16.48919140549705880072844381033, 17.51886587932505095099796449603, 18.53199415975255541366188073912, 19.24487827016131051726455938144, 20.925846540932275367319336643729, 22.09107117178282667772946933546, 22.39415393997094884308569729467, 23.770822512153994158763222639858, 24.11408355787692627341978736440, 24.790555902270009867627026265179, 26.14012045072168194302034715767

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