Properties

Label 1-373-373.161-r1-0-0
Degree $1$
Conductor $373$
Sign $0.374 + 0.927i$
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  + (−0.988 + 0.151i)2-s + (0.0506 + 0.998i)3-s + (0.954 − 0.299i)4-s + (−0.848 + 0.528i)5-s + (−0.201 − 0.979i)6-s + (0.250 + 0.968i)7-s + (−0.897 + 0.440i)8-s + (−0.994 + 0.101i)9-s + (0.758 − 0.651i)10-s + (−0.897 − 0.440i)11-s + (0.347 + 0.937i)12-s + (0.440 − 0.897i)13-s + (−0.394 − 0.918i)14-s + (−0.571 − 0.820i)15-s + (0.820 − 0.571i)16-s + (0.954 − 0.299i)17-s + ⋯
L(s)  = 1  + (−0.988 + 0.151i)2-s + (0.0506 + 0.998i)3-s + (0.954 − 0.299i)4-s + (−0.848 + 0.528i)5-s + (−0.201 − 0.979i)6-s + (0.250 + 0.968i)7-s + (−0.897 + 0.440i)8-s + (−0.994 + 0.101i)9-s + (0.758 − 0.651i)10-s + (−0.897 − 0.440i)11-s + (0.347 + 0.937i)12-s + (0.440 − 0.897i)13-s + (−0.394 − 0.918i)14-s + (−0.571 − 0.820i)15-s + (0.820 − 0.571i)16-s + (0.954 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8195264055 + 0.5528234031i\)
\(L(\frac12)\) \(\approx\) \(0.8195264055 + 0.5528234031i\)
\(L(1)\) \(\approx\) \(0.5977780137 + 0.2888880152i\)
\(L(1)\) \(\approx\) \(0.5977780137 + 0.2888880152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.988 + 0.151i)T \)
3 \( 1 + (0.0506 + 0.998i)T \)
5 \( 1 + (-0.848 + 0.528i)T \)
7 \( 1 + (0.250 + 0.968i)T \)
11 \( 1 + (-0.897 - 0.440i)T \)
13 \( 1 + (0.440 - 0.897i)T \)
17 \( 1 + (0.954 - 0.299i)T \)
19 \( 1 + (0.571 - 0.820i)T \)
23 \( 1 + (0.998 + 0.0506i)T \)
29 \( 1 + (0.688 + 0.724i)T \)
31 \( 1 + (-0.347 + 0.937i)T \)
37 \( 1 + (0.612 - 0.790i)T \)
41 \( 1 + (0.979 - 0.201i)T \)
43 \( 1 + (0.299 + 0.954i)T \)
47 \( 1 + (0.394 - 0.918i)T \)
53 \( 1 + (0.101 - 0.994i)T \)
59 \( 1 + (-0.688 - 0.724i)T \)
61 \( 1 + (0.998 - 0.0506i)T \)
67 \( 1 + (-0.848 + 0.528i)T \)
71 \( 1 + (0.954 + 0.299i)T \)
73 \( 1 + (-0.994 - 0.101i)T \)
79 \( 1 + (-0.651 - 0.758i)T \)
83 \( 1 + (-0.994 - 0.101i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.299 - 0.954i)T \)
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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

−24.16988461430571802117093934989, −23.60918475114521121073953133216, −22.94197999231295091915297410444, −20.92114273793719012687705003590, −20.63690143217609088473909417647, −19.643239899235830888227760094624, −18.90544958631242319945003702635, −18.29214515743119444938454413400, −17.0686279304844371234496359135, −16.66972505573015619164792766001, −15.568820515673431904098532226080, −14.397791610653809978970730150830, −13.21429371728748034175428160239, −12.33499526242302204843102899064, −11.547154847017449979258403644114, −10.69955299536170484325077464374, −9.4866267022292390272131400405, −8.299328200311202974059714737681, −7.69263710633403197759848871712, −7.11761917212897418759737811951, −5.80317256796189731154825292652, −4.18507415014834299530202763276, −2.91735387955862493659106598569, −1.47176045164460102255733899149, −0.73414263243805346815495553701, 0.63306268485636323132648188126, 2.79475499171624456126098787588, 3.17500345699783727432253684575, 5.06251936579496023326267748305, 5.79853615478499416325626978802, 7.28480778818032361716058263978, 8.235880280781882674226060511283, 8.871860998794103272467746601790, 10.0034776815300614192287948882, 10.92757757560877716124098285561, 11.378577116646034831214654840380, 12.5479298148167501274348751180, 14.387064203778278111857826090489, 15.08525219422806111854755597114, 15.94023167755595829134624171888, 16.143681102929971577360075310984, 17.69082187434603970923069316840, 18.32456917661340696069806322666, 19.24347497775469046632188604467, 20.07116169248009619376211390015, 21.02179731116640673786951010101, 21.68313775457946124054359190644, 22.89711685989943903997251845106, 23.64466208173790800902215853918, 24.90880685185916384911443828233

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