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Properties

Label 1-775-775.227-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.729 - 0.684i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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  + i·2-s + (0.406 − 0.913i)3-s − 4-s + (0.913 + 0.406i)6-s + (−0.406 + 0.913i)7-s − i·8-s + (−0.669 − 0.743i)9-s + (−0.978 − 0.207i)11-s + (−0.406 + 0.913i)12-s + (0.866 + 0.5i)13-s + (−0.913 − 0.406i)14-s + 16-s + (0.866 − 0.5i)17-s + (0.743 − 0.669i)18-s + (0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + i·2-s + (0.406 − 0.913i)3-s − 4-s + (0.913 + 0.406i)6-s + (−0.406 + 0.913i)7-s − i·8-s + (−0.669 − 0.743i)9-s + (−0.978 − 0.207i)11-s + (−0.406 + 0.913i)12-s + (0.866 + 0.5i)13-s + (−0.913 − 0.406i)14-s + 16-s + (0.866 − 0.5i)17-s + (0.743 − 0.669i)18-s + (0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.729 - 0.684i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.729 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06510955165 - 0.1646196349i\)
\(L(\frac12)\) \(\approx\) \(0.06510955165 - 0.1646196349i\)
\(L(1)\) \(\approx\) \(0.8533496931 + 0.1891135694i\)
\(L(1)\) \(\approx\) \(0.8533496931 + 0.1891135694i\)

Euler product

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

−22.45168711492677670614272727871, −21.26219572690814094943330720024, −20.8386183319405001255025032015, −20.32689087989506449605603061673, −19.41451662838554986955930385896, −18.69708869347015333207423925632, −17.72768192672076293291991373194, −16.7058866768304411505011239342, −16.13190227256502789660920494036, −14.907078331884244716064957166563, −14.29029182776953068073457061877, −13.21970297894051654409975845570, −12.87037913610818752540774863759, −11.47079027934521102041037068089, −10.65931011747537712794372145999, −10.140609715667505335002362971205, −9.536524146797606196972495122910, −8.23381675540665653026900356517, −7.86943527018465668081630075057, −6.05109767902329489571746528377, −5.08364518166285372741825882330, −4.10983085406044567019047013794, −3.44027219151214308075468681024, −2.628906090260673154672069643740, −1.24222851292293469589649742785, 0.04116828862837694603860942383, 1.291213541374719627028369206262, 2.7416378133980875369643110436, 3.53515782082065234366670557613, 5.161006903334671857760775611, 5.74062640205790372350624281037, 6.746016696516762648426915288215, 7.39928050685518705670665512522, 8.42212728894384144914738748769, 8.93370399408602028210759316282, 9.81332950604745088352605254892, 11.28029767420953690137316515583, 12.26106519247247582522608681659, 13.17305255962975355949350143801, 13.55787462368898884962987818545, 14.55461209874885050506254805311, 15.40869153258219442418464333772, 16.00569451404582960733165568961, 16.99314520239895592667672657469, 17.993893985537337819400567214615, 18.64882179714698309348076947558, 18.908302123141583673617709055020, 20.098405991706554671720548303053, 21.215930114905414345583328493420, 21.93107158569377855054095533395

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