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Properties

Label 1-385-385.214-r0-0-0
Degree $1$
Conductor $385$
Sign $0.746 + 0.665i$
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  + (0.104 + 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.809 − 0.587i)6-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)12-s + (0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (0.978 − 0.207i)18-s + (−0.978 − 0.207i)19-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)24-s + (0.669 + 0.743i)26-s + (0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.809 − 0.587i)6-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)12-s + (0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (0.978 − 0.207i)18-s + (−0.978 − 0.207i)19-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)24-s + (0.669 + 0.743i)26-s + (0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7969632941 + 0.3039086892i\)
\(L(\frac12)\) \(\approx\) \(0.7969632941 + 0.3039086892i\)
\(L(1)\) \(\approx\) \(0.7101941555 + 0.3918875518i\)
\(L(1)\) \(\approx\) \(0.7101941555 + 0.3918875518i\)

Euler product

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

−23.87840923010596320835541196497, −23.56176503091781830591471381550, −22.669275571183345668559682138261, −21.72306903764520596482503498372, −21.07475302300163317693411606136, −19.892865826896617475400550833978, −19.02056977094933926567189331522, −18.59087946472816341376157896311, −17.43062507508339843568405154551, −16.93952665803484413663773522449, −15.5094059641895415626508310566, −14.22415047423974613346105574434, −13.40878641394990241215539108156, −12.64807497596990106283359376559, −11.83564032010147290009394662931, −10.96566953128533720154736029603, −10.295281884630723911865046425478, −8.902870419472983127794852332207, −8.08121469273748552513887791775, −6.66064979122560116096572869856, −5.74392535716836395288955151759, −4.63714724523008816254005952533, −3.50014959440291844537428035080, −2.06412446333604644500550330562, −1.20595864230173805113394400267, 0.641070552006673849278949736650, 3.08994656667596580494770863740, 4.27004747564156954431225500009, 5.03856773661411502119734717008, 6.07594717945580718061044998647, 6.78637335726616960359187050683, 8.15298878063183782893733636393, 9.02292925339269055573087890415, 10.01879805713945168295962716506, 10.939375637721343271740936102466, 12.12467790357490722866996379541, 13.09852350067617216862003439507, 14.147909103799944107792328962239, 15.145715134365143795570396499298, 15.77992278588670740413717247605, 16.55041659124478070855760318749, 17.40444483040782863073617294085, 18.098236252853682158280314370428, 19.08086633729713784677323762808, 20.591023151013128236361352128300, 21.29192633403198344229327496988, 22.27286303146828373691981763004, 23.048552462163331064342182826605, 23.442708282123014869905522724697, 24.70761470992347083394592111149

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