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Properties

Label 1-373-373.19-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.236 - 0.971i$
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  + (0.897 + 0.440i)2-s + (−0.151 + 0.988i)3-s + (0.612 + 0.790i)4-s + (−0.101 + 0.994i)5-s + (−0.571 + 0.820i)6-s + (−0.688 + 0.724i)7-s + (0.201 + 0.979i)8-s + (−0.954 − 0.299i)9-s + (−0.528 + 0.848i)10-s + (0.201 − 0.979i)11-s + (−0.874 + 0.485i)12-s + (−0.979 − 0.201i)13-s + (−0.937 + 0.347i)14-s + (−0.968 − 0.250i)15-s + (−0.250 + 0.968i)16-s + (0.612 + 0.790i)17-s + ⋯
L(s)  = 1  + (0.897 + 0.440i)2-s + (−0.151 + 0.988i)3-s + (0.612 + 0.790i)4-s + (−0.101 + 0.994i)5-s + (−0.571 + 0.820i)6-s + (−0.688 + 0.724i)7-s + (0.201 + 0.979i)8-s + (−0.954 − 0.299i)9-s + (−0.528 + 0.848i)10-s + (0.201 − 0.979i)11-s + (−0.874 + 0.485i)12-s + (−0.979 − 0.201i)13-s + (−0.937 + 0.347i)14-s + (−0.968 − 0.250i)15-s + (−0.250 + 0.968i)16-s + (0.612 + 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.236 - 0.971i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.236 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9873710941 + 1.256629746i\)
\(L(\frac12)\) \(\approx\) \(-0.9873710941 + 1.256629746i\)
\(L(1)\) \(\approx\) \(0.7421516062 + 1.092827285i\)
\(L(1)\) \(\approx\) \(0.7421516062 + 1.092827285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.897 + 0.440i)T \)
3 \( 1 + (-0.151 + 0.988i)T \)
5 \( 1 + (-0.101 + 0.994i)T \)
7 \( 1 + (-0.688 + 0.724i)T \)
11 \( 1 + (0.201 - 0.979i)T \)
13 \( 1 + (-0.979 - 0.201i)T \)
17 \( 1 + (0.612 + 0.790i)T \)
19 \( 1 + (0.968 - 0.250i)T \)
23 \( 1 + (-0.988 + 0.151i)T \)
29 \( 1 + (-0.758 - 0.651i)T \)
31 \( 1 + (0.874 + 0.485i)T \)
37 \( 1 + (-0.918 + 0.394i)T \)
41 \( 1 + (0.820 + 0.571i)T \)
43 \( 1 + (0.790 - 0.612i)T \)
47 \( 1 + (0.937 + 0.347i)T \)
53 \( 1 + (0.299 + 0.954i)T \)
59 \( 1 + (0.758 + 0.651i)T \)
61 \( 1 + (-0.988 - 0.151i)T \)
67 \( 1 + (-0.101 + 0.994i)T \)
71 \( 1 + (0.612 - 0.790i)T \)
73 \( 1 + (-0.954 + 0.299i)T \)
79 \( 1 + (-0.848 - 0.528i)T \)
83 \( 1 + (-0.954 + 0.299i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.790 + 0.612i)T \)
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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

−23.82686450258585548457043373476, −22.716340347334308034046718317444, −22.5410713092558960988924128752, −20.93983314303185441060990771890, −20.123111376872769622301461895154, −19.755517193663424344035349015130, −18.76786628287364675624871307815, −17.52822014097600466653930772268, −16.61825117435560711502307722307, −15.81991532672118070269443869358, −14.35851373972094449732133264679, −13.79021328802831561208461094886, −12.74368183115514814100305800408, −12.30238198878091144845259420939, −11.58697088294541005123697155191, −10.09501643515356571311672860399, −9.36623961046634773250035946095, −7.58949846194454316565168823335, −7.09405037279554478983748525657, −5.81952433638405261470192696492, −4.93076108135117514431752921607, −3.84793449459948342195210391774, −2.49353661504076551957643576576, −1.37233093220426992735598280760, −0.320895408287316208017632986260, 2.627472695484269525543122004605, 3.24224470125638351388377604766, 4.17324774373131518238317853805, 5.68386137775243454814800493177, 5.948171779507372018162099624709, 7.26856407616769655309988625317, 8.41052294254648149447083009806, 9.66635285501177151143085756604, 10.58717834187184882623699829766, 11.650102633654602530084583789343, 12.24396969214410594276010582860, 13.7411546423300309928100627136, 14.41795442432619728959442729302, 15.31488627873996925634121139673, 15.82219619942056567070588182633, 16.75994510019118071215721449614, 17.675335203628344385986291755914, 19.042432804332058891061505601522, 19.84278814263546668018932220913, 21.11335699195653130435170139877, 21.90498532934267480504721929795, 22.2452625015922250515382713718, 22.92978537758662752579700134237, 24.0685153716994675615982861405, 24.99606604478409957685811827203

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