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Properties

Label 1-385-385.233-r0-0-0
Degree $1$
Conductor $385$
Sign $0.0931 + 0.995i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
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  + (−0.994 − 0.104i)2-s + (−0.743 + 0.669i)3-s + (0.978 + 0.207i)4-s + (0.809 − 0.587i)6-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.866 + 0.5i)12-s + (−0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (0.994 − 0.104i)17-s + (−0.207 + 0.978i)18-s + (−0.978 + 0.207i)19-s + (0.866 − 0.5i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.743 + 0.669i)3-s + (0.978 + 0.207i)4-s + (0.809 − 0.587i)6-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.866 + 0.5i)12-s + (−0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (0.994 − 0.104i)17-s + (−0.207 + 0.978i)18-s + (−0.978 + 0.207i)19-s + (0.866 − 0.5i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0931 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0931 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.0931 + 0.995i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.0931 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3981786336 + 0.3626576272i\)
\(L(\frac12)\) \(\approx\) \(0.3981786336 + 0.3626576272i\)
\(L(1)\) \(\approx\) \(0.5274585561 + 0.1495964781i\)
\(L(1)\) \(\approx\) \(0.5274585561 + 0.1495964781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (0.406 + 0.913i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−24.51147111181561184075170909244, −23.523145906802088316616706225226, −22.830046966728033057259162463557, −21.582620977446094971071840462324, −20.72106898735596819665321963043, −19.34901568568768999035537707125, −19.1839304472595175614493323991, −18.00881680879029271189322128887, −17.30360529633143063708009009865, −16.77661196278829191978952774928, −15.6525545180903150054356177809, −14.78236309944635952158436048570, −13.39970045430899526797220368339, −12.345446475024192911457445993613, −11.67728657527581795054092236349, −10.57774287584319522780705048556, −9.98494889353688694808754751072, −8.57092315077908682344359058479, −7.74447784849179534452791467421, −6.89251264895267618746792598099, −5.95126638926083640011234389743, −4.986018645059176249777397178296, −3.02227162412402831434032816726, −1.82301871801282038535392823508, −0.5769056701632867815119562918, 1.12580777614410676324658017431, 2.65478583627317211939472769746, 3.96335716816424072276363421630, 5.20481546543926413934792461674, 6.36852418539237585706776452132, 7.16474241338692174316234812565, 8.49706687988617399370016380591, 9.38619076795491865593758142660, 10.22865236048114177129082173277, 10.956232207561523602518133380451, 11.94212277485120165722121649037, 12.581553253185703882051890252546, 14.39460017500035421466056079965, 15.20762625493232004778679859352, 16.22122148533811346543120241575, 16.8695978024357029608789472779, 17.460947876158752048605627834222, 18.606748853244200490091318011093, 19.22947213522420965035694461788, 20.4441770627486905047287893187, 21.19910956054254886138006072696, 21.81404844854427869940333379773, 23.0375854347047061005964621626, 23.835988420218811699984280632495, 24.84707088498526100767704211832

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