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Properties

Label 1-373-373.170-r0-0-0
Degree $1$
Conductor $373$
Sign $0.999 + 0.0167i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.283 − 0.959i)2-s + (−0.713 + 0.701i)3-s + (−0.839 − 0.543i)4-s + (−0.985 + 0.168i)5-s + (0.470 + 0.882i)6-s + (−0.954 − 0.299i)7-s + (−0.758 + 0.651i)8-s + (0.0168 − 0.999i)9-s + (−0.117 + 0.993i)10-s + (0.943 − 0.331i)11-s + (0.979 − 0.201i)12-s + (−0.758 − 0.651i)13-s + (−0.557 + 0.830i)14-s + (0.585 − 0.810i)15-s + (0.409 + 0.912i)16-s + (−0.0506 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.283 − 0.959i)2-s + (−0.713 + 0.701i)3-s + (−0.839 − 0.543i)4-s + (−0.985 + 0.168i)5-s + (0.470 + 0.882i)6-s + (−0.954 − 0.299i)7-s + (−0.758 + 0.651i)8-s + (0.0168 − 0.999i)9-s + (−0.117 + 0.993i)10-s + (0.943 − 0.331i)11-s + (0.979 − 0.201i)12-s + (−0.758 − 0.651i)13-s + (−0.557 + 0.830i)14-s + (0.585 − 0.810i)15-s + (0.409 + 0.912i)16-s + (−0.0506 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.999 + 0.0167i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.999 + 0.0167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5746710812 + 0.004812234855i\)
\(L(\frac12)\) \(\approx\) \(0.5746710812 + 0.004812234855i\)
\(L(1)\) \(\approx\) \(0.6219210533 - 0.1629294701i\)
\(L(1)\) \(\approx\) \(0.6219210533 - 0.1629294701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.283 - 0.959i)T \)
3 \( 1 + (-0.713 + 0.701i)T \)
5 \( 1 + (-0.985 + 0.168i)T \)
7 \( 1 + (-0.954 - 0.299i)T \)
11 \( 1 + (0.943 - 0.331i)T \)
13 \( 1 + (-0.758 - 0.651i)T \)
17 \( 1 + (-0.0506 + 0.998i)T \)
19 \( 1 + (-0.994 + 0.101i)T \)
23 \( 1 + (-0.250 + 0.968i)T \)
29 \( 1 + (0.990 - 0.134i)T \)
31 \( 1 + (0.979 + 0.201i)T \)
37 \( 1 + (0.780 + 0.625i)T \)
41 \( 1 + (0.528 - 0.848i)T \)
43 \( 1 + (-0.839 - 0.543i)T \)
47 \( 1 + (0.997 - 0.0675i)T \)
53 \( 1 + (0.0168 + 0.999i)T \)
59 \( 1 + (-0.378 + 0.925i)T \)
61 \( 1 + (-0.713 + 0.701i)T \)
67 \( 1 + (0.347 - 0.937i)T \)
71 \( 1 + (0.890 + 0.455i)T \)
73 \( 1 + (0.857 - 0.514i)T \)
79 \( 1 + (-0.117 + 0.993i)T \)
83 \( 1 + (0.0168 + 0.999i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.0506 + 0.998i)T \)
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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

−24.769850838083334381158903312512, −23.63201562614265035509426689845, −23.01923693024811479959090980141, −22.43043496245681730511779298199, −21.61457133901232428415200793738, −19.8830786827736336629731044838, −19.16338476301878773638605847470, −18.421402635464000309449936305069, −17.278701024383222848015061512418, −16.537976893090968297754837390999, −15.98246954441882651073372186555, −14.88501675801671695470670428823, −13.94520366422249394151553491567, −12.72760547215614372677914126138, −12.27482880007729088881058945065, −11.46767976744980289005043621857, −9.828878317805689920976269082404, −8.77549612419648561921860607246, −7.74694042568299781400766777601, −6.68966065256609863739870449192, −6.44548870333944171966293037433, −4.88750133049623748193754646158, −4.20590701711825624891649780296, −2.67334252436521179064331385014, −0.50722832546779406351342197673, 0.86904867399169105137410148632, 2.919625831009994316178243425002, 3.84851219696907241534124957594, 4.401129617547848918875734044331, 5.7932545873124756124103611878, 6.69957926986864654163144870600, 8.33268742549826114092851116824, 9.43305176685139777877672687902, 10.336808355911581456837535759555, 10.94063211656495992061148244836, 12.13674055240247594480638356386, 12.36723268671871826416956539973, 13.74493159991874769064432436321, 15.00322515690680178378074399440, 15.484385952317814672135389560037, 16.814140507777669182407013293155, 17.40710852943827720020525962325, 18.7797628508518001161849173227, 19.66694715540339454312214182992, 19.952639006004355741099126281022, 21.37434189999777644135755122257, 22.00509718473127857640781113387, 22.76124320664048604697200947864, 23.31916644444596001900281946827, 24.13934252486126349698191353893

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