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Properties

Label 1-671-671.169-r0-0-0
Degree $1$
Conductor $671$
Sign $0.550 + 0.834i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (−0.978 − 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (0.669 + 0.743i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (0.669 + 0.743i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.550 + 0.834i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.550 + 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4371747485 + 0.2353237069i\)
\(L(\frac12)\) \(\approx\) \(0.4371747485 + 0.2353237069i\)
\(L(1)\) \(\approx\) \(0.5393450232 + 0.03041894710i\)
\(L(1)\) \(\approx\) \(0.5393450232 + 0.03041894710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−22.49786838321451588192534718179, −21.52402082062691047823318427952, −21.24825875733332357679852004559, −20.09758865957034253382916076285, −19.338770502413266642216564975657, −18.21214806213227094718589728617, −17.57172788758307854915065594811, −17.06661490890113177652722984064, −16.1662049329934712525283630451, −15.49480503371901772948062251365, −14.81541316450541557482012094444, −13.36301554574197882642781137827, −12.18267977919273666372011218233, −11.74863050633034953786937795080, −10.61766227731578534473747135358, −9.84369543216108384948730395168, −9.03203169331574980717309653175, −8.63584323020753737890269744541, −6.96370921767376315506053711544, −6.35165356426561477118568666606, −5.21884503253041674748221751010, −4.81870593972244924542227672080, −2.85564973712802082395538662923, −1.852007191750829808174652905929, −0.407002292417227034881574178170, 1.122059504516587037616847777869, 2.10457858311819966032832084724, 3.14317092467569317107278746328, 4.674007089453217159133929499298, 6.01703790772230751460715551789, 6.78360419832976996439493539435, 7.29537397260408553652615819851, 8.27876392066446090785657743639, 9.673112880772557071671979185179, 10.35824799950758502455016986340, 10.868716294561758318660267987879, 11.79577956815785835036759979555, 12.762419814098827599117129731874, 13.56943732632094249877137971260, 14.638755188495237986780338466349, 15.73424576498875334497278230625, 16.859734539746387983287575084822, 17.25864244402029064600181607634, 17.82008102131055663590568727645, 18.798212381523553164594641740311, 19.3180876775683668557611031080, 20.267854806416960967441564050769, 21.24049583089567211211094286223, 22.13659486761594926356036271239, 22.7913958906328732015116500279

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