Properties

Degree $2$
Conductor $147$
Sign $0.749 + 0.661i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 − 0.358i)2-s + (0.5 − 0.866i)3-s + (0.914 − 1.58i)4-s + (1.70 + 2.95i)5-s − 0.414·6-s − 1.58·8-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (1 − 1.73i)11-s + (−0.914 − 1.58i)12-s − 2.58·13-s + 3.41·15-s + (−1.49 − 2.59i)16-s + (−1.12 + 1.94i)17-s + (−0.207 + 0.358i)18-s + (−1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.146 − 0.253i)2-s + (0.288 − 0.499i)3-s + (0.457 − 0.791i)4-s + (0.763 + 1.32i)5-s − 0.169·6-s − 0.560·8-s + (−0.166 − 0.288i)9-s + (0.223 − 0.387i)10-s + (0.301 − 0.522i)11-s + (−0.263 − 0.457i)12-s − 0.717·13-s + 0.881·15-s + (−0.374 − 0.649i)16-s + (−0.271 + 0.471i)17-s + (−0.0488 + 0.0845i)18-s + (−0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
Motivic weight: \(1\)
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21112 - 0.457928i\)
\(L(\frac12)\) \(\approx\) \(1.21112 - 0.457928i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.207 + 0.358i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.70 - 2.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 + (1.12 - 1.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.82 - 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + (0.585 - 1.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.585 - 1.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.12 - 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.82 + 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 + (-6.94 + 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.82 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + (7.12 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−13.14047675504863346171648618576, −11.62050185568625122740774259941, −10.93508584965005132476859004473, −9.976648794703119148991022507653, −9.078299705054228347854104539727, −7.32111794603211418314867556170, −6.55030564562380442777133054919, −5.56402701137154873391531483432, −3.11633590144782128325846682748, −1.93060194382655682929945983018, 2.30619245105572922245043296140, 4.17526723950096684770236833405, 5.35204718184660538430895470686, 6.84111233223846220194210314734, 8.136709774759497825112418332544, 9.035351245279866829599741745252, 9.762587299902956233208971903441, 11.20430087055643746324755995282, 12.52093737635660074777789998122, 12.84605580745188520116095561807

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