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Properties

Label 1-373-373.20-r1-0-0
Degree $1$
Conductor $373$
Sign $0.440 + 0.897i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (−0.968 + 0.250i)2-s + (−0.820 + 0.571i)3-s + (0.874 − 0.485i)4-s + (−0.394 − 0.918i)5-s + (0.651 − 0.758i)6-s + (0.994 + 0.101i)7-s + (−0.724 + 0.688i)8-s + (0.347 − 0.937i)9-s + (0.612 + 0.790i)10-s + (−0.724 − 0.688i)11-s + (−0.440 + 0.897i)12-s + (−0.688 + 0.724i)13-s + (−0.988 + 0.151i)14-s + (0.848 + 0.528i)15-s + (0.528 − 0.848i)16-s + (0.874 − 0.485i)17-s + ⋯
L(s)  = 1  + (−0.968 + 0.250i)2-s + (−0.820 + 0.571i)3-s + (0.874 − 0.485i)4-s + (−0.394 − 0.918i)5-s + (0.651 − 0.758i)6-s + (0.994 + 0.101i)7-s + (−0.724 + 0.688i)8-s + (0.347 − 0.937i)9-s + (0.612 + 0.790i)10-s + (−0.724 − 0.688i)11-s + (−0.440 + 0.897i)12-s + (−0.688 + 0.724i)13-s + (−0.988 + 0.151i)14-s + (0.848 + 0.528i)15-s + (0.528 − 0.848i)16-s + (0.874 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.440 + 0.897i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4855952251 + 0.3026450142i\)
\(L(\frac12)\) \(\approx\) \(0.4855952251 + 0.3026450142i\)
\(L(1)\) \(\approx\) \(0.5000113417 + 0.06572487302i\)
\(L(1)\) \(\approx\) \(0.5000113417 + 0.06572487302i\)

Euler product

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

−24.162683333404921911820462789661, −23.551682179295918968332576563354, −22.46431530826908416914621578485, −21.631962091751344740042467914641, −20.57579340579959787293189131399, −19.62117232345007513295324702909, −18.62244641814581580839791149359, −18.20103129252702727810223686341, −17.36752508262967156787414083681, −16.69049410152275556846579176191, −15.31010743597184773380667855207, −14.78454536503800670869563288316, −13.13290749236686618707177284905, −12.15887771585019626130757219667, −11.46438550481151362822403224275, −10.49469212563657443975568797135, −10.14634269950693903952691971309, −8.20521783555644787326980564078, −7.69756171446673648547081605389, −6.912257257307861280562167384358, −5.77486392294317049158994106999, −4.431424791860137910465279746757, −2.72204016304356186362732637063, −1.83350311132229984917183628035, −0.3751444876603901185867917190, 0.7150317519556424634742781268, 1.945175685645355576440036413994, 3.867767960672605585782785478376, 5.19465573485915290534974051801, 5.62216950395021644867404371507, 7.16784188195675907925926545010, 8.101847062197535676548341919205, 8.96345641424702445684950556946, 9.9419457554340365533440303998, 10.89607043153218161767448315157, 11.73829880983931190583902204744, 12.322227956924103471196569083421, 14.07314095306890133964337367363, 15.17757133675446913092717085667, 15.92554021304124370053802795262, 16.775014865443990253172390365596, 17.19162119159514999306975257525, 18.28086097767171198083657493210, 19.06237345272395871630355199067, 20.23896017409867774019924834141, 21.15038041315276750990969888304, 21.441332453752756795124551532218, 23.19346039482774486857797312962, 23.88912334178294445628312032367, 24.34242944437768892116164755565

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