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Properties

Label 1-667-667.206-r0-0-0
Degree $1$
Conductor $667$
Sign $0.341 + 0.939i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.433 + 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.781 − 0.623i)10-s + (0.974 − 0.222i)11-s + i·12-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.433 + 0.900i)15-s + (−0.222 − 0.974i)16-s + i·17-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.781 − 0.623i)10-s + (0.974 − 0.222i)11-s + i·12-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.433 + 0.900i)15-s + (−0.222 − 0.974i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.341 + 0.939i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.341 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448773913 + 1.014605518i\)
\(L(\frac12)\) \(\approx\) \(1.448773913 + 1.014605518i\)
\(L(1)\) \(\approx\) \(1.261965840 + 0.5126558305i\)
\(L(1)\) \(\approx\) \(1.261965840 + 0.5126558305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
3 \( 1 + (0.781 - 0.623i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.974 - 0.222i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.781 + 0.623i)T \)
31 \( 1 + (0.433 + 0.900i)T \)
37 \( 1 + (0.974 + 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.781 + 0.623i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.433 - 0.900i)T \)
79 \( 1 + (-0.974 - 0.222i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (-0.781 - 0.623i)T \)
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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

−22.51425494395760614339722093381, −21.85625635111060128312801119577, −20.77792597807871413606815617, −20.17235779938880758772383983763, −19.658344918607483787829703549777, −18.94092223939519205127255592583, −18.0433815863760851818209435160, −16.63324887022198730846652076959, −15.55229520176226787010895505413, −15.326198549707112411831517609857, −14.27019441871903480443189782362, −13.37026891675324444292643086252, −12.54964098101486613715891750807, −11.74580899509708667257614807039, −10.98938079089641090744051422744, −9.72606870409610745007785075025, −9.2635012920067396303988096994, −8.48513158307856442425959458172, −7.33961004154480871553822165568, −5.81143366725373135975726848275, −4.866310330761058416301425693169, −4.00897456725948188079101406197, −3.18125738133148457974034848856, −2.46543672200743665590556603477, −0.86036845130599718024641307316, 1.13877879361464869487699720997, 2.91758935389216278240474384664, 3.86099127823308835080559950631, 4.13707489126431149495237344126, 6.04984859142776982086146352958, 6.79040459024031447497966109368, 7.31207134051242963946111470820, 8.25622782088107303115563923992, 8.983733369391908103016700080869, 10.05101830527459291489156292573, 11.56152822470227344367682908093, 12.24930701796219880328619297522, 13.17332555148179481130583397931, 14.0619970908434271952413030201, 14.46872124859524782626989553190, 15.38219483301223334104367950793, 16.29095034711495428471884538167, 16.94758345942228433014197751976, 18.068461920414479635085237221, 18.95400180906099178053058431637, 19.530743091793804248883365083982, 20.34549115254026173157835361108, 21.482751032356122621529071766, 22.38870213997677690513287633784, 23.23748316899057299792340464136

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