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Properties

Label 1-373-373.158-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.141 - 0.989i$
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  + (0.250 + 0.968i)2-s + (0.820 − 0.571i)3-s + (−0.874 + 0.485i)4-s + (−0.918 + 0.394i)5-s + (0.758 + 0.651i)6-s + (−0.994 − 0.101i)7-s + (−0.688 − 0.724i)8-s + (0.347 − 0.937i)9-s + (−0.612 − 0.790i)10-s + (−0.688 + 0.724i)11-s + (−0.440 + 0.897i)12-s + (0.688 − 0.724i)13-s + (−0.151 − 0.988i)14-s + (−0.528 + 0.848i)15-s + (0.528 − 0.848i)16-s + (−0.874 + 0.485i)17-s + ⋯
L(s)  = 1  + (0.250 + 0.968i)2-s + (0.820 − 0.571i)3-s + (−0.874 + 0.485i)4-s + (−0.918 + 0.394i)5-s + (0.758 + 0.651i)6-s + (−0.994 − 0.101i)7-s + (−0.688 − 0.724i)8-s + (0.347 − 0.937i)9-s + (−0.612 − 0.790i)10-s + (−0.688 + 0.724i)11-s + (−0.440 + 0.897i)12-s + (0.688 − 0.724i)13-s + (−0.151 − 0.988i)14-s + (−0.528 + 0.848i)15-s + (0.528 − 0.848i)16-s + (−0.874 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.141 - 0.989i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.141 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2248961494 - 0.2594565450i\)
\(L(\frac12)\) \(\approx\) \(0.2248961494 - 0.2594565450i\)
\(L(1)\) \(\approx\) \(0.7699653075 + 0.1835041256i\)
\(L(1)\) \(\approx\) \(0.7699653075 + 0.1835041256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.250 + 0.968i)T \)
3 \( 1 + (0.820 - 0.571i)T \)
5 \( 1 + (-0.918 + 0.394i)T \)
7 \( 1 + (-0.994 - 0.101i)T \)
11 \( 1 + (-0.688 + 0.724i)T \)
13 \( 1 + (0.688 - 0.724i)T \)
17 \( 1 + (-0.874 + 0.485i)T \)
19 \( 1 + (-0.528 - 0.848i)T \)
23 \( 1 + (-0.820 - 0.571i)T \)
29 \( 1 + (-0.954 - 0.299i)T \)
31 \( 1 + (-0.440 - 0.897i)T \)
37 \( 1 + (-0.0506 + 0.998i)T \)
41 \( 1 + (-0.758 - 0.651i)T \)
43 \( 1 + (0.874 - 0.485i)T \)
47 \( 1 + (-0.151 + 0.988i)T \)
53 \( 1 + (-0.347 - 0.937i)T \)
59 \( 1 + (-0.954 - 0.299i)T \)
61 \( 1 + (-0.820 + 0.571i)T \)
67 \( 1 + (-0.918 + 0.394i)T \)
71 \( 1 + (-0.874 - 0.485i)T \)
73 \( 1 + (0.347 + 0.937i)T \)
79 \( 1 + (0.612 + 0.790i)T \)
83 \( 1 + (0.347 + 0.937i)T \)
89 \( 1 + T \)
97 \( 1 + (0.874 - 0.485i)T \)
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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

−24.790684422852130807404248335060, −23.72465840199131385318692465441, −23.013336942587886631961115501828, −21.97803782937188074408742945987, −21.28090428660602642128399757588, −20.37021355238494868171560418649, −19.75015208786350008267834249801, −19.0087387606170353996419130969, −18.37154897999806140512306965891, −16.48561139661626895202785508420, −15.95757679122608198419904222991, −15.0304990707715193973067749902, −13.85692704120196422637776023078, −13.22210776436198336877602236669, −12.3031498793185503430750330858, −11.16377036338376847610904562431, −10.44341606260461171521892879423, −9.203899505156486555429046429804, −8.778508345155930633377864165025, −7.64705091219894619039735317041, −5.96207160932323875590947896134, −4.67774134369640298741689747388, −3.73568764151732084956720362572, −3.187466857850222776195832055862, −1.84977016693535548807209261837, 0.1637378722337591385343743983, 2.53705323641816717066027854129, 3.56857750952647468051811634279, 4.37138161129643577363354150110, 6.0807246649469693454441346300, 6.856614557691964295474848064545, 7.67731989225811114315303213001, 8.42255717029511859235963703924, 9.411693861869811921090523560018, 10.623121250421697009148601071799, 12.220755291429395509826691446749, 13.00547659019477900057400769155, 13.50644113202749141809259133754, 14.8611383890686479290388232022, 15.388145627233178126347318446949, 15.960330295865263992279304866403, 17.336524089277120509811705528290, 18.311059709630925136110243341017, 18.92847694593613977597041716173, 19.91314068699450837144293714935, 20.686987684489126481607860977594, 22.24104606650436572824758484360, 22.74337070032153720638513146698, 23.82295333164963073128342849027, 24.1118866217723192839361998124

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