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Properties

Label 1-373-373.181-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.912 + 0.408i$
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  + (0.557 + 0.830i)2-s + (0.780 + 0.625i)3-s + (−0.378 + 0.925i)4-s + (−0.585 + 0.810i)5-s + (−0.0843 + 0.996i)6-s + (0.688 − 0.724i)7-s + (−0.979 + 0.201i)8-s + (0.217 + 0.975i)9-s + (−0.999 − 0.0337i)10-s + (0.315 + 0.948i)11-s + (−0.874 + 0.485i)12-s + (0.979 + 0.201i)13-s + (0.985 + 0.168i)14-s + (−0.963 + 0.266i)15-s + (−0.713 − 0.701i)16-s + (−0.612 − 0.790i)17-s + ⋯
L(s)  = 1  + (0.557 + 0.830i)2-s + (0.780 + 0.625i)3-s + (−0.378 + 0.925i)4-s + (−0.585 + 0.810i)5-s + (−0.0843 + 0.996i)6-s + (0.688 − 0.724i)7-s + (−0.979 + 0.201i)8-s + (0.217 + 0.975i)9-s + (−0.999 − 0.0337i)10-s + (0.315 + 0.948i)11-s + (−0.874 + 0.485i)12-s + (0.979 + 0.201i)13-s + (0.985 + 0.168i)14-s + (−0.963 + 0.266i)15-s + (−0.713 − 0.701i)16-s + (−0.612 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.912 + 0.408i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.912 + 0.408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4243702822 + 1.987153427i\)
\(L(\frac12)\) \(\approx\) \(0.4243702822 + 1.987153427i\)
\(L(1)\) \(\approx\) \(1.039732324 + 1.251140158i\)
\(L(1)\) \(\approx\) \(1.039732324 + 1.251140158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.557 + 0.830i)T \)
3 \( 1 + (0.780 + 0.625i)T \)
5 \( 1 + (-0.585 + 0.810i)T \)
7 \( 1 + (0.688 - 0.724i)T \)
11 \( 1 + (0.315 + 0.948i)T \)
13 \( 1 + (0.979 + 0.201i)T \)
17 \( 1 + (-0.612 - 0.790i)T \)
19 \( 1 + (0.250 + 0.968i)T \)
23 \( 1 + (-0.151 - 0.988i)T \)
29 \( 1 + (-0.184 + 0.982i)T \)
31 \( 1 + (-0.874 - 0.485i)T \)
37 \( 1 + (-0.801 - 0.598i)T \)
41 \( 1 + (0.820 + 0.571i)T \)
43 \( 1 + (0.378 - 0.925i)T \)
47 \( 1 + (-0.638 - 0.769i)T \)
53 \( 1 + (-0.217 + 0.975i)T \)
59 \( 1 + (0.943 - 0.331i)T \)
61 \( 1 + (-0.780 - 0.625i)T \)
67 \( 1 + (0.994 + 0.101i)T \)
71 \( 1 + (0.990 + 0.134i)T \)
73 \( 1 + (0.736 + 0.676i)T \)
79 \( 1 + (0.999 + 0.0337i)T \)
83 \( 1 + (0.217 - 0.975i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.612 + 0.790i)T \)
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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

−24.13389518909169265577148165911, −23.67334308989760022915507103479, −22.38322796012501214469044758214, −21.20092599797700315791298740746, −20.912449079669591589811373525911, −19.62780411417554579689025932528, −19.45062630749370170502925782191, −18.33234606326153040995294781022, −17.536349267312463513341932238489, −15.777976947188449833309808505241, −15.23383742131737709160179944343, −14.123182658065568124255297847821, −13.322214987903977775744081580344, −12.65106474554135423209316817387, −11.596702821532628821331877830198, −11.10872142407499690946687555462, −9.304258712728254315017869007284, −8.73759503470957192186606519317, −7.96641035915893969303073228586, −6.32682265984798716269453430606, −5.34129408382523296221675831506, −4.07420146167722701617308852892, −3.26164859839033812422121033199, −1.9428169587823031129569318145, −1.01887030759431760244197830990, 2.18919173044318674351740964297, 3.6162576172725941001924902286, 4.08216922814155424946913390233, 5.095781574172353392278784936305, 6.68285762569842107377009295799, 7.439975541532793008218648567185, 8.216869825765027887451672920898, 9.241379422301488114916194896910, 10.518909752246016848135107593906, 11.37372800098365863903564362341, 12.635018516152979597521184133740, 13.915456530062293827961923762152, 14.32965997988531072698160176753, 15.06916950732150609067917459825, 15.932756206540792629478671731300, 16.66269984126358425311926412760, 17.95041077957625686612331438276, 18.630433110552773341846914682, 20.18374329666483962408512858508, 20.53230282660813520040655361098, 21.62013143368871226214371484269, 22.64163493332625293020355817563, 23.05786341090500721755385486891, 24.161042378712784642593926943413, 25.05105665328489821003411381730

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