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Properties

Label 1-373-373.43-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.552 + 0.833i$
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  + (−0.0337 + 0.999i)2-s + (0.184 − 0.982i)3-s + (−0.997 − 0.0675i)4-s + (0.543 + 0.839i)5-s + (0.975 + 0.217i)6-s + (−0.918 − 0.394i)7-s + (0.101 − 0.994i)8-s + (−0.931 − 0.363i)9-s + (−0.857 + 0.514i)10-s + (−0.912 − 0.409i)11-s + (−0.250 + 0.968i)12-s + (0.994 − 0.101i)13-s + (0.425 − 0.905i)14-s + (0.925 − 0.378i)15-s + (0.990 + 0.134i)16-s + (0.440 + 0.897i)17-s + ⋯
L(s)  = 1  + (−0.0337 + 0.999i)2-s + (0.184 − 0.982i)3-s + (−0.997 − 0.0675i)4-s + (0.543 + 0.839i)5-s + (0.975 + 0.217i)6-s + (−0.918 − 0.394i)7-s + (0.101 − 0.994i)8-s + (−0.931 − 0.363i)9-s + (−0.857 + 0.514i)10-s + (−0.912 − 0.409i)11-s + (−0.250 + 0.968i)12-s + (0.994 − 0.101i)13-s + (0.425 − 0.905i)14-s + (0.925 − 0.378i)15-s + (0.990 + 0.134i)16-s + (0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4892343309 + 0.9117464050i\)
\(L(\frac12)\) \(\approx\) \(0.4892343309 + 0.9117464050i\)
\(L(1)\) \(\approx\) \(0.8376684361 + 0.2853711965i\)
\(L(1)\) \(\approx\) \(0.8376684361 + 0.2853711965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.0337 + 0.999i)T \)
3 \( 1 + (0.184 - 0.982i)T \)
5 \( 1 + (0.543 + 0.839i)T \)
7 \( 1 + (-0.918 - 0.394i)T \)
11 \( 1 + (-0.912 - 0.409i)T \)
13 \( 1 + (0.994 - 0.101i)T \)
17 \( 1 + (0.440 + 0.897i)T \)
19 \( 1 + (0.790 - 0.612i)T \)
23 \( 1 + (-0.651 - 0.758i)T \)
29 \( 1 + (-0.985 - 0.168i)T \)
31 \( 1 + (0.250 + 0.968i)T \)
37 \( 1 + (0.664 - 0.747i)T \)
41 \( 1 + (-0.954 + 0.299i)T \)
43 \( 1 + (-0.0675 + 0.997i)T \)
47 \( 1 + (0.996 + 0.0843i)T \)
53 \( 1 + (0.363 + 0.931i)T \)
59 \( 1 + (-0.638 + 0.769i)T \)
61 \( 1 + (0.982 + 0.184i)T \)
67 \( 1 + (-0.998 + 0.0506i)T \)
71 \( 1 + (0.557 + 0.830i)T \)
73 \( 1 + (0.780 + 0.625i)T \)
79 \( 1 + (-0.514 - 0.857i)T \)
83 \( 1 + (-0.931 + 0.363i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.897 + 0.440i)T \)
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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

−23.84079209971773327537611157325, −22.84216620481358928848877662610, −22.164486199058590700307602086554, −21.29391760750094235250996620067, −20.4409599275978512501791014279, −20.30493406133939216078870655532, −18.85541616079806804261159408877, −18.151668045722881363746670758421, −16.917013936612993510639866451306, −16.16744102597407076835743609634, −15.31622611893559298101943012545, −13.82477871039218285940637750622, −13.38277575820761992772504198716, −12.28510294857064102181757212496, −11.399735741514317075131472889986, −10.08934290829874826364423492110, −9.72179936317599098593718552253, −8.92115639083745379546925620768, −7.93071237745409026620230803424, −5.74283251481694616701534854241, −5.23850154528523414215418082745, −3.98123121353181623754382931708, −3.09526879747881212544347860126, −1.95224339331141170299133876138, −0.32849569238586447935035054173, 1.05588153127478146280139485334, 2.7746197216842668715134176267, 3.70506887150741185043556145615, 5.65140741887269492681499919891, 6.167139806716754961034612408166, 7.04308749107776860866492601383, 7.85884287271074655585270977647, 8.86980799876978434367658044072, 10.00750431041854722233176553882, 10.928991101570569493907199292900, 12.56595180162227082305534506878, 13.38549757437585188814002074556, 13.80435856044002290426061480755, 14.80610380939147738078199328203, 15.84272898613065300194300030030, 16.74825394467108007741389032901, 17.78323522445711059305821118922, 18.44908844987903665734792847654, 18.97241693382530284555562627289, 20.069032448559948292894999120407, 21.458231671501031157428718037621, 22.44413227436461059719158922917, 23.20232350858947497931790756913, 23.76584375951409814194310399698, 24.8013623301458118692479214342

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