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Properties

Label 1-373-373.101-r0-0-0
Degree $1$
Conductor $373$
Sign $0.882 + 0.471i$
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  + (0.585 − 0.810i)2-s + (0.470 + 0.882i)3-s + (−0.315 − 0.948i)4-s + (−0.931 − 0.363i)5-s + (0.990 + 0.134i)6-s + (0.347 + 0.937i)7-s + (−0.954 − 0.299i)8-s + (−0.557 + 0.830i)9-s + (−0.839 + 0.543i)10-s + (0.736 + 0.676i)11-s + (0.688 − 0.724i)12-s + (−0.954 + 0.299i)13-s + (0.963 + 0.266i)14-s + (−0.117 − 0.993i)15-s + (−0.801 + 0.598i)16-s + (0.979 + 0.201i)17-s + ⋯
L(s)  = 1  + (0.585 − 0.810i)2-s + (0.470 + 0.882i)3-s + (−0.315 − 0.948i)4-s + (−0.931 − 0.363i)5-s + (0.990 + 0.134i)6-s + (0.347 + 0.937i)7-s + (−0.954 − 0.299i)8-s + (−0.557 + 0.830i)9-s + (−0.839 + 0.543i)10-s + (0.736 + 0.676i)11-s + (0.688 − 0.724i)12-s + (−0.954 + 0.299i)13-s + (0.963 + 0.266i)14-s + (−0.117 − 0.993i)15-s + (−0.801 + 0.598i)16-s + (0.979 + 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.882 + 0.471i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.882 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.562765838 + 0.3912514835i\)
\(L(\frac12)\) \(\approx\) \(1.562765838 + 0.3912514835i\)
\(L(1)\) \(\approx\) \(1.363014177 + 0.01843686712i\)
\(L(1)\) \(\approx\) \(1.363014177 + 0.01843686712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.585 - 0.810i)T \)
3 \( 1 + (0.470 + 0.882i)T \)
5 \( 1 + (-0.931 - 0.363i)T \)
7 \( 1 + (0.347 + 0.937i)T \)
11 \( 1 + (0.736 + 0.676i)T \)
13 \( 1 + (-0.954 + 0.299i)T \)
17 \( 1 + (0.979 + 0.201i)T \)
19 \( 1 + (0.918 - 0.394i)T \)
23 \( 1 + (0.528 + 0.848i)T \)
29 \( 1 + (0.0168 + 0.999i)T \)
31 \( 1 + (0.688 + 0.724i)T \)
37 \( 1 + (0.0843 - 0.996i)T \)
41 \( 1 + (-0.612 + 0.790i)T \)
43 \( 1 + (-0.315 - 0.948i)T \)
47 \( 1 + (-0.713 - 0.701i)T \)
53 \( 1 + (-0.557 - 0.830i)T \)
59 \( 1 + (0.857 - 0.514i)T \)
61 \( 1 + (0.470 + 0.882i)T \)
67 \( 1 + (0.151 + 0.988i)T \)
71 \( 1 + (-0.664 - 0.747i)T \)
73 \( 1 + (0.997 - 0.0675i)T \)
79 \( 1 + (-0.839 + 0.543i)T \)
83 \( 1 + (-0.557 - 0.830i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.979 + 0.201i)T \)
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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

−24.4144173600486282773150324819, −23.86869185894684408196224749200, −22.88860467493160015008292193753, −22.466097192367205458101495226859, −20.95017117143839183219433796866, −20.17659161042692425334916903344, −19.19616936998551931117793648796, −18.43275112508386726227866314703, −17.214765684269432502184858068291, −16.6869979589354359114401215213, −15.42337661490185355442261864744, −14.41703625583504685234040996862, −14.15888888490137228035935693037, −13.04537230160657070181408122099, −12.00679949662371128839103188763, −11.47092314295122007602640915797, −9.780871531885460792599701084440, −8.332030892316765487474089686214, −7.77324458334492786774737965937, −7.08354214571614979222668432335, −6.176522149198289704353677595676, −4.73490197602651500794632665665, −3.62647150422549795959292998027, −2.87774498197264666683837968890, −0.824173908598807729777396912366, 1.60273025306390026155516123240, 2.92888448541191760461320657532, 3.74376735503926477321011461095, 4.892502540298322081293377236637, 5.28887422404448604183156725536, 7.16111180754880375000693441448, 8.50380195273168641326903331829, 9.31702903485822991038720685239, 10.06062131913876772582849009042, 11.43937759153345281590553721359, 11.86808410010968183128913910411, 12.76971247897288543891053963435, 14.21713852403390800840932487911, 14.810225952514317483397206969260, 15.45568184564329385815634849105, 16.43060996433047295669320504580, 17.72976003710814831143741610780, 19.06450061605164550028049213408, 19.61310301750871222233402977604, 20.32594276709852781451437600976, 21.23920503309435166377563002954, 21.92278772597971582508946220000, 22.66898801817066770857719465805, 23.62298595102616077359625796072, 24.63855985720257238651289702775

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