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Properties

Label 1-373-373.369-r0-0-0
Degree $1$
Conductor $373$
Sign $0.106 - 0.994i$
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  + (−0.999 − 0.0337i)2-s + (−0.184 + 0.982i)3-s + (0.997 + 0.0675i)4-s + (−0.839 + 0.543i)5-s + (0.217 − 0.975i)6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (−0.931 − 0.363i)9-s + (0.857 − 0.514i)10-s + (0.409 − 0.912i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (−0.905 − 0.425i)14-s + (−0.378 − 0.925i)15-s + (0.990 + 0.134i)16-s + (−0.440 − 0.897i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0337i)2-s + (−0.184 + 0.982i)3-s + (0.997 + 0.0675i)4-s + (−0.839 + 0.543i)5-s + (0.217 − 0.975i)6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (−0.931 − 0.363i)9-s + (0.857 − 0.514i)10-s + (0.409 − 0.912i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (−0.905 − 0.425i)14-s + (−0.378 − 0.925i)15-s + (0.990 + 0.134i)16-s + (−0.440 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.106 - 0.994i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.106 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1552407453 - 0.1395610818i\)
\(L(\frac12)\) \(\approx\) \(0.1552407453 - 0.1395610818i\)
\(L(1)\) \(\approx\) \(0.4558832202 + 0.1073382866i\)
\(L(1)\) \(\approx\) \(0.4558832202 + 0.1073382866i\)

Euler product

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

−24.68713539634598505181409967992, −24.129049674242143147312633818869, −23.52103496649065011064009220077, −22.354453291128852132995148496803, −20.84076621717942782248183041948, −20.10622925377662137773272762534, −19.52414160987752840052360583512, −18.69459436472368087417162862184, −17.52893211479482321506058384810, −17.26947913510835890837592082887, −16.29241415387458904656471085325, −14.97582993410848142065656105691, −14.3933730975263241913473824538, −12.54763193652976349330327707137, −12.29640376615643198354317599660, −11.241992688055082350701696962563, −10.39671907043972650517557178878, −8.95920827051755732404284197975, −8.10893159379390423349265923683, −7.50861642124277318958141027037, −6.684077303941461616768511372007, −5.301940853763550418034530905754, −3.99269154969687106113986984820, −2.15002834159403306339934248246, −1.41215827067488717051119420994, 0.17667401818265206841795039691, 2.25940218876166088681938217801, 3.33200425530100983165427924865, 4.54823817252939076868010440081, 5.77172031495140251007272740015, 7.0121256729143435528718414405, 8.03932771224214911501617886622, 8.873694777627219651257812631306, 9.75348622843986588173796271115, 10.93676069681199399541936310342, 11.41406812868649598407987970114, 12.006642101443167901350079652471, 14.070981203659101336022034344585, 15.08540274909090261022515428083, 15.460306721891807933212259137615, 16.53566429571628255899084129114, 17.26952205498641412630479206192, 18.21630900373038133441800852188, 19.153430438703186600902838117396, 19.95214476896298233069158339186, 20.77215540682530954948143714788, 21.826250088762521857532362923701, 22.28906094155896536732140124701, 23.84872608021832295999181447639, 24.33751610536132114137691663866

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