Properties

Degree $2$
Conductor $147$
Sign $0.968 - 0.250i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (0.500 + 0.866i)4-s + (−1 + 1.73i)5-s + 0.999·6-s + 3·8-s + (−0.499 + 0.866i)9-s + (0.999 + 1.73i)10-s + (−2 − 3.46i)11-s + (−0.499 + 0.866i)12-s + 2·13-s − 1.99·15-s + (0.500 − 0.866i)16-s + (−3 − 5.19i)17-s + (0.499 + 0.866i)18-s + (2 − 3.46i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 + 0.433i)4-s + (−0.447 + 0.774i)5-s + 0.408·6-s + 1.06·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.603 − 1.04i)11-s + (−0.144 + 0.249i)12-s + 0.554·13-s − 0.516·15-s + (0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (0.117 + 0.204i)18-s + (0.458 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.968 - 0.250i$
Motivic weight: \(1\)
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43298 + 0.182610i\)
\(L(\frac12)\) \(\approx\) \(1.43298 + 0.182610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.35538967567012999742132355678, −11.75036984020560182464243259419, −11.19055537214842297399625531298, −10.46146611385686836717395235055, −8.986821541781670224412933437681, −7.84630499662072424196307054559, −6.79861725246347443155001270562, −4.98665985961532829464182856583, −3.53248899785197925579273264205, −2.75783616986689583278023535180, 1.77125131081290015271079329676, 4.13814467201104539154440020783, 5.37487154127442046284491509799, 6.56478569624224668215301513432, 7.66247519775145860938089500169, 8.525092380382120147497930696518, 9.950414922461296240777473019533, 11.03457210395954380294007920053, 12.37706777544349875402830591154, 13.02330501545202800489756566632

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