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Properties

Label 1-373-373.60-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.997 + 0.0651i$
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  + (0.810 + 0.585i)2-s + (−0.470 − 0.882i)3-s + (0.315 + 0.948i)4-s + (0.363 − 0.931i)5-s + (0.134 − 0.990i)6-s + (−0.347 − 0.937i)7-s + (−0.299 + 0.954i)8-s + (−0.557 + 0.830i)9-s + (0.839 − 0.543i)10-s + (−0.676 + 0.736i)11-s + (0.688 − 0.724i)12-s + (0.954 − 0.299i)13-s + (0.266 − 0.963i)14-s + (−0.993 + 0.117i)15-s + (−0.801 + 0.598i)16-s + (−0.979 − 0.201i)17-s + ⋯
L(s)  = 1  + (0.810 + 0.585i)2-s + (−0.470 − 0.882i)3-s + (0.315 + 0.948i)4-s + (0.363 − 0.931i)5-s + (0.134 − 0.990i)6-s + (−0.347 − 0.937i)7-s + (−0.299 + 0.954i)8-s + (−0.557 + 0.830i)9-s + (0.839 − 0.543i)10-s + (−0.676 + 0.736i)11-s + (0.688 − 0.724i)12-s + (0.954 − 0.299i)13-s + (0.266 − 0.963i)14-s + (−0.993 + 0.117i)15-s + (−0.801 + 0.598i)16-s + (−0.979 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.997 + 0.0651i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.997 + 0.0651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01176237712 - 0.3609042154i\)
\(L(\frac12)\) \(\approx\) \(0.01176237712 - 0.3609042154i\)
\(L(1)\) \(\approx\) \(1.089056818 - 0.1252793920i\)
\(L(1)\) \(\approx\) \(1.089056818 - 0.1252793920i\)

Euler product

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

−24.78849952246379819506671192822, −23.47036691423517213292701428346, −22.9278291146845110027485804988, −22.09288433618859804107581766922, −21.347007284272397688156990203467, −21.08884146780413976954731729089, −19.6584940405711597180889880360, −18.66235851651224254722909840419, −18.118508252163095689763960192306, −16.62654547707306558846458009038, −15.50368731655617596390881303839, −15.25890143689628608190813919263, −14.082205124558607634356150414483, −13.231899123702487026777788040906, −12.07107532980643844213118377358, −11.06750921398209632058703471402, −10.702852812821060573451676387608, −9.63301471977632951427969658466, −8.69026968648669575947156583770, −6.656458857046825633887098443748, −5.939033069504661686467717961195, −5.271856509144962335962129106331, −3.81289129652642373020523875409, −3.12738848119544232638316526893, −1.97582635455223751679594944853, 0.074125855304306918602248714371, 1.52666423252787812770144018637, 2.877002444652714881147966648390, 4.47635474676507534169508295589, 5.10311418680181935744110060871, 6.32042022874267897146192057303, 6.97795012504469395170231977978, 7.99827562749222643058495875019, 8.93288139958187960726303268487, 10.580923270059276273909171840615, 11.49548265129956941596187857234, 12.77070915758134688198817698445, 13.167029783931347233526732946017, 13.60227064913584113434183353938, 15.02712125775642784780126449970, 16.12512406352162379658033567186, 16.79331980236937135578797298343, 17.56741807842946775168716764300, 18.29492440367985976171333436333, 19.99355735966750198677533550278, 20.335056222462533361355441003141, 21.46074945773287783840787769861, 22.57942973526396336962464389902, 23.25014842767371003235530011159, 23.9068680649207535241630037788

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