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Properties

Label 1-147-147.47-r0-0-0
Degree $1$
Conductor $147$
Sign $0.886 - 0.462i$
Analytic cond. $0.682665$
Root an. cond. $0.682665$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (−0.0747 + 0.997i)11-s + (0.900 − 0.433i)13-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 + 0.781i)20-s + (0.623 − 0.781i)22-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (−0.0747 + 0.997i)11-s + (0.900 − 0.433i)13-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 + 0.781i)20-s + (0.623 − 0.781i)22-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8710740534 - 0.2135615665i\)
\(L(\frac12)\) \(\approx\) \(0.8710740534 - 0.2135615665i\)
\(L(1)\) \(\approx\) \(0.8524863310 - 0.1736436859i\)
\(L(1)\) \(\approx\) \(0.8524863310 - 0.1736436859i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (-0.365 - 0.930i)T \)
59 \( 1 + (0.955 + 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
97 \( 1 - T \)
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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

−28.3604438090366345511813179705, −26.973244773153388861882114547114, −26.33046522523681804750862594263, −25.42975487968468822379603364870, −24.53276650656432612318589178631, −23.66656933962049414769237559300, −22.335864793937917772120964128491, −21.27345581313370856362837279528, −20.18106505296821508289133936740, −18.927447573360848177021011619508, −18.21459755777650057933432852666, −17.27654999213501611176914264294, −16.30879596799621530564954231687, −15.306479807749359083144485794875, −14.03353498331441299769409272749, −13.35402686625603249397259899462, −11.28596080916438156940479833448, −10.63627512237271258444693714225, −9.24957276724889659406853093226, −8.64435353999156886976335998899, −7.00934538584630663993299287761, −6.20663552302135401180994557150, −5.05770389474620820703819284965, −2.86663095056923174095153061969, −1.3381310091934355626126419186, 1.395020838893411210088971658255, 2.55160647187024820802091897866, 4.183018575452513773746174822520, 5.84931426833977307846649169016, 7.17344352831192470362868354275, 8.47931433564995084203379509598, 9.485150924881992745153188596735, 10.31853679404343999738597711453, 11.454988650653885681256064130042, 12.78102009913089235413056633874, 13.43542754653002490671215553095, 15.10521369420137208233874661117, 16.306922196428561388447014739716, 17.39683020780692134964496334598, 17.97632930714710268290510089149, 19.025635527793111159794684636564, 20.47483346536900564700141765241, 20.71814773615794081717436144596, 21.97178285281184551513916509235, 22.94130128430194458307765777610, 24.635939352105343414569345111406, 25.329470672119905760381197056083, 26.15114366196109008548627490421, 27.212126831203919461067985616273, 28.42591746279868865837521226020

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