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Properties

Label 1-373-373.73-r0-0-0
Degree $1$
Conductor $373$
Sign $0.975 - 0.219i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.943 + 0.331i)2-s + (0.283 − 0.959i)3-s + (0.780 + 0.625i)4-s + (0.857 + 0.514i)5-s + (0.585 − 0.810i)6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (−0.839 − 0.543i)9-s + (0.638 + 0.769i)10-s + (0.470 + 0.882i)11-s + (0.820 − 0.571i)12-s + (0.528 − 0.848i)13-s + (−0.315 − 0.948i)14-s + (0.736 − 0.676i)15-s + (0.217 + 0.975i)16-s + (0.151 − 0.988i)17-s + ⋯
L(s)  = 1  + (0.943 + 0.331i)2-s + (0.283 − 0.959i)3-s + (0.780 + 0.625i)4-s + (0.857 + 0.514i)5-s + (0.585 − 0.810i)6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (−0.839 − 0.543i)9-s + (0.638 + 0.769i)10-s + (0.470 + 0.882i)11-s + (0.820 − 0.571i)12-s + (0.528 − 0.848i)13-s + (−0.315 − 0.948i)14-s + (0.736 − 0.676i)15-s + (0.217 + 0.975i)16-s + (0.151 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.975 - 0.219i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.975 - 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.748635544 - 0.3049565181i\)
\(L(\frac12)\) \(\approx\) \(2.748635544 - 0.3049565181i\)
\(L(1)\) \(\approx\) \(2.105046438 - 0.1140437392i\)
\(L(1)\) \(\approx\) \(2.105046438 - 0.1140437392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.943 + 0.331i)T \)
3 \( 1 + (0.283 - 0.959i)T \)
5 \( 1 + (0.857 + 0.514i)T \)
7 \( 1 + (-0.612 - 0.790i)T \)
11 \( 1 + (0.470 + 0.882i)T \)
13 \( 1 + (0.528 - 0.848i)T \)
17 \( 1 + (0.151 - 0.988i)T \)
19 \( 1 + (-0.954 + 0.299i)T \)
23 \( 1 + (0.688 - 0.724i)T \)
29 \( 1 + (-0.117 + 0.993i)T \)
31 \( 1 + (0.820 + 0.571i)T \)
37 \( 1 + (-0.557 - 0.830i)T \)
41 \( 1 + (-0.994 - 0.101i)T \)
43 \( 1 + (0.780 + 0.625i)T \)
47 \( 1 + (-0.664 - 0.747i)T \)
53 \( 1 + (-0.839 + 0.543i)T \)
59 \( 1 + (-0.801 - 0.598i)T \)
61 \( 1 + (0.283 - 0.959i)T \)
67 \( 1 + (-0.874 + 0.485i)T \)
71 \( 1 + (-0.931 - 0.363i)T \)
73 \( 1 + (0.890 + 0.455i)T \)
79 \( 1 + (0.638 + 0.769i)T \)
83 \( 1 + (-0.839 + 0.543i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.151 - 0.988i)T \)
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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

−24.61371881327855112687393620559, −23.73745765420800323237079746074, −22.52323002689092289457852616082, −21.84189229288607029429872466560, −21.250045040003376045459557877874, −20.78907551677811897601929757931, −19.350744985918114763315302356571, −19.089119080920500853613153952985, −17.162369839565340190422881648, −16.50555404574439313727010923514, −15.564997418257778597950067304907, −14.843160163799208123891035260, −13.74420701416935535928407805766, −13.2531099145135212033088524253, −12.0295245470707709497027445770, −11.120905100271530143909347510143, −10.11000437658288279908184384409, −9.24147687719877709548085832997, −8.48959727989854638417199798263, −6.313697395797089663465987332370, −5.94109019961132573058068550078, −4.80131318814265460131050670478, −3.816643242088164018962753345, −2.812560319127885151903376706762, −1.69243286358787578544540028791, 1.48543974486123265184213922597, 2.68807944000684404713557415114, 3.48999152080686800066360286172, 4.976050912427369282104794492629, 6.23454108377438477755720759074, 6.77932063731455511589493961203, 7.50743536653342413051963978058, 8.82296621734184783844848375723, 10.185014274209507273316527721156, 11.12680503427531242114591184684, 12.539947580524719772524755303993, 12.90392205715540013814455852393, 13.9234129223935398684175135068, 14.387649066354911634165047794996, 15.410597647407372535704507337273, 16.73889286227474059848561216161, 17.43040232429245472418153237550, 18.289717322018211176915931544485, 19.48545828149446566781692230351, 20.37227384957901750619038338753, 20.980968404208323125237547924313, 22.40427847552278667664603192508, 22.90162540210079361534835307742, 23.47135149979713275906786121207, 24.8267949336507916413752895930

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