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Properties

Label 1-147-147.122-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: $\chi_{147} (122, \cdot )$
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

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

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